False positives (and even true negatives) have non-zero cost.

http://articles.mercola.com/sites/articles/archive/2013/03/0...

False positives can have a ton of very negative effects. There is the monetary cost of further testing, which can be high depending on what is being tested for, and most importantly, telling a bunch of people that they have some particular disease when they don't is really bad. They may become more stressed, which can bring about a whole host of problems, people might start telling family members, and start getting their affairs in order, they might make different financial decisions based on these false results. And its actually very difficult to administer these tests in a way that doesn't results in way way more false positives than true positives.

Arguably this attitude is a contributor the problem of antibiotic resistance.