But the euphoria has begun to fade. Investment bankers caution that the San Francisco company might be unable to go public at $10 billion, much less deliver a big pop to recent investors and employees who hoped to strike it rich, according to people familiar with the matter.

BlackRock Inc., which led the $350 million deal that more than doubled Dropbox’s valuation, has cut its estimate of the company’s per-share value by 24%, securities filings show.

Dropbox responds that it is continuing to increase its business, added 500 employees in the past year, including senior executives, and has no need for additional capital from private or public investors.

Still, the company is a portent of wider trouble for startups that found it easy to attract money at sky’s-the-limit valuations in the continuing technology boom. The market for initial public offerings has turned chilly and inhospitable, largely because technology companies have sought valuations above what public investors are willing to pay.

So far this year, only 14% of IPOs in the U.S. were done by tech companies, the smallest percentage since at least the mid-1990s, according to Dealogic.

Many U.S. based companies that went public this year have seen their stock prices suffer, posting a median return of zero compared with their IPO price. Investors who bought shares after the IPOs began trading, often the first chance many individual investors get, are down by a median of 13%.

And as of Friday, at least 11 of 49 venture-capital-backed U.S. technology companies with IPOs since the start of 2014 traded below the per-share value where they last raised money as a private company, an analysis of stock-sale documents by The Wall Street Journal shows.

The data suggest that even some of the most promising startups in Silicon Valley might be worth far less in the eyes of the rest of the investment world. The risk is that the lackluster reception for tech startups in the stock market could ricochet through companies that are still private.

If that happens, it would likely be harder to recruit and pay employees. It could even threaten the future pipeline of startup funding.

Stark change

Venture capitalist Bill Gurley, a partner at Benchmark, says he saw a stark change in the third quarter. “We’re seeing financing rounds where founders are coming back and lowering the price over and over again,” he says.

For example, online marketplace Jet.com Inc. has told investors as part of a fundraising effort begun in the past two weeks that it is aiming for $500 million and a valuation of $2 billion, say people familiar with the matter.

This summer, Jet said privately that it would be seeking a $3 billion valuation, these people say. “We don’t comment on financing and have never discussed valuation publicly,” a Jet spokesman said.

In one way, Dropbox suffers from being compared with rival online storage firm Box Inc., which went public in December, according to investment bankers. Box has a stock-market value of $1.65 billion, or less than six times its expected fiscal 2016 revenue. Dropbox would need revenue of roughly $2 billion to support a similar valuation in the stock market, but one investment banker estimates the company’s revenue is likely to hit just $500 million this year. Dropbox declines to comment on its revenue.

“The thing that worries me the most about all these [billion-dollar valuations] is that you need a public market to get liquid,” says Chris Douvos, managing director of Venture Investment Associates, a Peapack, N.J., firm that invests in funds and startups. “But who’s going to buy at these valuations? It’s all priced for perfection.”

Cupps Capital Management LLC bought shares of On Deck Capital Inc. when the online-lending company went public in December. The $20-a-share IPO price was about one-third higher than the price of On Deck’s last private-share sale earlier in 2014.

Since then, the stock price has tumbled to about $9. Andrew Cupps, the chief investment officer of Cupps Capital Management, says he escaped with a small profit by selling the Chicago investment firm’s stake last spring. “The company hasn’t really disappointed, but I do think that the investor base is doing a gut check on just how differentiated it will be” from traditional banks, he says.

At least 124 private companies are valued by venture-capital firms at $1 billion or more, nearly double the number a year ago, according to Dow Jones VentureSource. Bidding wars for tech startups are so common that 10 companies raised capital at billion-dollar valuations in September alone.

But those investors could have trouble cashing in if the IPO market isn’t able to support even higher valuations. A lower valuation as a private or public company also can sap the morale of startup employees who endure pressure and all-nighters in return for the possibility of a big payday.

In a sign of wariness among pre-IPO investors, an analysis of funding rounds by law firm Fenwick & West LLP in March found that 30% of private companies valued at $1 billion or more promised a specified IPO price. In some cases, the companies agreed to give additional equity to investors if the IPO price wasn’t met.

Some venture capitalists and executives at startup companies aren’t worried. They say startups have never been cheaper to build and are maturing faster than fledgling companies did in the last tech boom, partly because the smartphone era creates an easily reachable market of two billion people.

As a result, closely held startups such as Dropbox, car-hailing service Uber Technologies Inc. and home-sharing service Airbnb Inc. each have annual revenue of hundreds of millions of dollars. Fast-growing revenue streams help young companies continue to attract investors and stay private longer.

That same magnetic appeal is pushing valuations higher and higher. Some of the pressure comes from mutual funds, pension funds and other money managers who are hungry for higher returns than they can earn elsewhere. Rock-bottom interest rates have increased that appetite.

More than two-thirds of venture-capital-backed private companies in the U.S. valued at $1 billion or more are backed by a public-market investor such as a big mutual fund, hedge fund or bank, according to Dow Jones VentureSource. Those investors usually are counting on an IPO.

That strategy worked for Fidelity Investments after it bought a stake in software maker Hubspot Inc. in late 2012 for $16.86 a share. Hubspot went public last October at $25 and closed Monday at nearly $50.

Earlier this year, Airbnb sought to raise funding at a valuation of $18 billion, according to a person familiar with the matter. Airbnb eventually convinced investors that included mutual-fund firms Wellington Management, Baillie Gifford, T. Rowe Price Group Inc. and Fidelity to invest $1.5 billion at a valuation of $25.5 billion. T. Rowe Price and Fidelity already owned Airbnb shares.

Airbnb is now the world’s third-highest-valued startup, behind Uber and smartphone maker Xiaomi Corp. Airbnb hasn’t publicly discussed any IPO plans, but many outsiders saw its July hiring of a new finance chief as a step in that direction. An Airbnb spokeswoman declined to comment.

Dropbox received the $10 billion-valuation offer from BlackRock after just two days of investor meetings, said a person familiar with the fundraising.

Mutual funds and hedge funds are also pushing up valuations for some companies in their earlier stages. OfferUp Inc., a peer-to-peer marketplace, raised about $75 million from T. Rowe Price and hedge-fund firms Coatue Management LLC and Tiger Global Management LLC.

The infusion in March valued four-year-old OfferUp, which has no revenue, at about $800 million, up 10 times from its first round of funding last year, say people familiar with the matter. Investors are betting OfferUp can take a bite out of ad-listings service Craigslist. A spokeswoman for OfferUp didn’t respond to multiple requests for comment.

Earlier this year, Zenefits Inc. got cold calls from investors willing to value it at $2 billion out of thin air, says Parker Conrad, co-founder and chief executive of the two-year-old San Francisco company. Zenefits gives away software to small businesses to help them automate human-resources chores, collecting a commission when employees sign up for health insurance and fees for other services offered through the software.

ENLARGE

The company’s valuation grew after Zenefits provided “an extensive virtual data room” of financial information on a hard drive to serious investors, said David Sacks, the company’s operating chief.

Zenefits has projected an annual revenue run rate of $100 million by January, or the equivalent of $8 million a month. In January 2015, Zenefits had monthly sales of about $1.6 million, according to the company.

Potential investors, including billionaire David Bonderman of TPG Capital, got a one-hour sales pitch from Mr. Conrad. “We can be a $100 billion company, giving investors today a huge return,” Mr. Conrad said. “We make a lot of money on insurance.”

Venture-capital firm Sequoia Capital decided not to invest in Zenefits.

Within a few days, three investors made separate offers that valued Zenefits at $3 billion or more. Mr. Conrad was about to sign a deal with a $3.5 billion valuation when his mentor, Lars Dalgaard, a general partner at venture-capital firm Andreessen Horowitz, urged waiting until the end of the day. Mr. Dalgaard later called it a “feeding frenzy.”

An hour later, Fidelity offered to make an investment that valued Zenefits at $4.5 billion, a staggering nine times the price investors paid just 11 months earlier. “We could have raised infinite amounts with a $3 billion valuation,” Mr. Conrad says.

Most of the investors willing to fund Zenefits at that valuation still said yes to $4.5 billion, including TPG and actors Ashton Kutcher and Jared Leto. Venture capitalist Vinod Khosla invested after Mr. Conrad called him and cited all the reasons “why you can still 10x your money,” says the Zenefits CEO. The deal was completed in May.

Relentless scrutiny

In private funding rounds, a company only has to convince one investor to set the valuation and then brings in other investors. Prices usually are based on multiyear financial projections. In contrast, startups that go public face relentless scrutiny of their results and forecasts. Their stock prices fluctuate daily.

Among the nine U.S.-listed IPOs since 2014 by venture-backed technology companies that were valued at $1 billion in private fundraising rounds and have since reported annual results, only three have met or exceeded analysts’ consensus profit forecast for that year, according to FactSet.

Since going public in December, online-lending marketplace LendingClub Corp., located 10 blocks from Zenefits in the trendy SoMa district of San Francisco, has seen its valuation shrink from $8.5 billion after its first day of trading to $5.4 billion. The share price has fallen amid concern that competition and regulation threaten LendingClub’s business model of matching borrowers with lenders.

New York City firefighter Brian Gitman bought 250 shares of LendingClub during the IPO and held on to them as the price slid. He regrets it.

Knowing that big-name investors put money into LendingClub when it was private gave Mr. Gitman, 33 years old, a false sense of security, he says. “It feels like that should be like the bumper in bowling,” he says.

Henry Ellenbogen, who manages T. Rowe Price’s technology investments, says the gap between what private and public investors are willing to pay is starting to narrow. “When times are good and valuations are strong, people think they have liquidity in the private markets, but that is illusory,” he says.

At Dropbox, the valuation dilemma looms large. Drew Houston, the company’s chief executive, rebuffed a takeover offer in 2009 by Apple Inc. co-founder Steve Jobs. The size of the offer wasn’t disclosed. Dropbox now has 400 million registered users, who share and store files through its Web service.

Most of them use Dropbox for free, though the company says more than 130,000 businesses pay at least $900 a year for a minimum of five employees. Customers include News Corp, which publishes The Wall Street Journal.

Online storage is fast becoming commoditized, and Dropbox now competes with Apple, Google Inc., Microsoft Corp. and Amazon.com Inc., the world’s four largest tech companies by stock-market value.

Dropbox says it is expanding its business beyond data storage. Last week, the company released a document-collaboration tool called Paper.

“Others can chatter about valuations,” a Dropbox spokeswoman said in a statement. “We’re hard at work building an enduring business.”

Write to Rolfe Winkler at rolfe.winkler@wsj.com, Douglas MacMillan at douglas.macmillan@wsj.com, Telis Demos at telis.demos@wsj.com and Monica Langley at monica.langley@wsj.com