Both results show that evaluating two tests on the same family of data will lead to a ~10% chance that a researcher will claim a “significant” result if they look for either test to reject the null. Any claim there is a maximum 5% false positive rate would be mistaken. As an exercise, verify that doing the same on \(m=4\) tests will lead to an ~18% chance!
A bad testing platform would be one that claims a maximum 5% false positive rate when any one of multiple tests on the same family of data show significance at the 5% level. Clearly, if a researcher is going to claim that the FWER is no more than \(\alpha\), then they must control for the FWER and carefully consider how individual tests reject the null.
This is worth taking some time to read carefully. H/T R-Bloggers