OK, I think I might be getting it. You can tell me if this sounds right.
This is the example I was thinking of (from the wiki you linked on alpha errors).
Perhaps the most widely discussed false positives in medical screening come from the breast cancer screening procedure mammography. The US rate of false positive mammograms is up to 15%, the highest in world. One consequence of the high false positive rate in the US is that, in any 10-year period, half of the American women screened receive a false positive mammogram. False positive mammograms are costly, with over $100 million spent annually in the U.S. on follow-up testing and treatment. They also cause women unneeded anxiety. As a result of the high false positive rate in the US, as many as 90–95% of women who get a positive mammogram do not have the condition. The lowest rate in the world is in the Netherlands, 1%. The lowest rates are generally in Northern Europe where mammography films are read twice and a high threshold for additional testing is set (the high threshold decreases the power of the test)
The false positives are higher in the US than in Northern Europe, not because of the equipment or procedure used, but because of the stringency of the test.
Because of that, false positives might be high for a certain set of data where fMRI technology is used but not for all data where fMRI technology is used.