One issue of concern is the possible cost of these false positives. If doctors are very careful to communicate to patients that 75% of positive results will only require a second test, then that's OK. Sadly, that doesn't seem to be what happens in many cases. The communication given to the patient is 'this is a test for disease X', and the patient rightfully interprets a positive result as 'I have X'.

The problem comes in because 'I have depression' could lead the patient to take risky or dangerous action. Sadly, misunderstanding on the side of doctors (who, at least the ones I know well, aren't given a strong education in statistics at university) might also lead them to make poor decisions. As long as everybody is honest about the meaning of the test, and understands the differences between P(Y), P(X), P(Y|X) and P(X|Y) then that's great. Sadly, there is good evidence that this is not always the case.

For an example more along HN's normal lines, consider hard drive monitoring data from SMART. Several public studies of SMART attributes in modern drives have shown that positive predictive value of SMART data as a drive failure indicator is very poor. Many of the drives which are replaced because they are failing SMART are actually just fine, but a lot of money, labor and downtime is spent on the incorrect assumption that P(healthy|SMART failure) is much smaller than P(unhealthy|SMART failure).