The Bonferroni inequality is used to determine an upper bound on the p-value of an experiment in which multiple hypotheses tests are performed, each hypothesis test of the Null hypothesis against its alternative resulting in a p-value. A common mistake is to take the p-value of the combined multiple hypothesis experiment as the smallest p-value pmin. However, the true p-value is bounded above by the smallest p-value pmin times N, the number of multiple hypothesis tests performed.
The upper bound Npmin is obtained by the Bonferroni inequality. The Bonferroni upper bound applies whether or not the various multiple hypothesis are dependent. However, if the correlations between the multiplie hypothesis are all positive, then the Bonferroni upper bound can be way too high.