The difficulty here is the nature of art itself. Because art relies so much upon perception, it is difficult to even define what art is, and what it is not. Every definition will lead you to problems. Consider the basic act of creation. We could try to make the argument that at the very least, art requires that someone created it. But what about a painting done by an elephant? Or a robot? What if it is unknown what created the piece? There are countless similar caveats in establishing a definition.

Some might consider the right approach to be to first establish an agreeable definition of art, and then from there, to set about answering the question. However, in doing so, you are no longer talking about the entirety of art, but a subset of what might be considered art. Of course, there is nothing wrong with settling the question within an agreed upon context, but it doesn't really settle the mysterious aspect of the question, and isn't entirely satisfying in the case of art. For many subjects, workable definitions are easier to agree upon. But art defies definition.

Oddly, if we agree upon a context for what we consider to be art, and then we might say that to the extent that art can be communicated, it might be taught. However, even then, there is negative space to communication that can lead us to understand things that we were never taught, and evidence of this kind of understanding arises in art all the time.

There can be little doubt that some people are more prone than others to spend their time and energy in a creative space, and that some are more receptive to expression than others. But art does not only arise from those that set out to create it, and for those that do, it doesn't always reveal itself as intended.

So my answer is: Art can and cannot be taught.