It seems to be a surprise to most people (certainly to me) how sharp the Bonferroni correction is when the number of tests is large. Unless the correlation between tests is really, high, the actual family-wise Type I error rate is very close to the nominal rate $$\alpha/k$$.
“In working on the various confidence intervals for $$k$$means, I thought of the Bonferroni inequality ones quite early, but since they were so simple I thought they couldn’t possibly be of any use. I spent a long time trying to prove that the confidence intervals which would be used in the case of independent variables could also be used or dependent variables. After failing to find a general proof for this, I finally noticed that the simple Bonferroni intervals were nearly as short”.