Take mammograms. It's a fairly non-invasive test for breast cancer. Everyone wants to reduce breast cancer, right? Unfortunately, the data is very much like fivethirtyeight's example. In a 10 year period, 1 in 2 women are harmed by a false positive and 1 in 5 were harmed by an unnecessary surgical procedure. What about the lives saved? None.[0] Screening mammograms only end up harming patients.
While it may make sense intuitively that more screening must be better, the data generally fails to back it up.
[0] http://www.thennt.com/nnt/screening-mammography-for-reducing...