When testing multiple hypotheses simultaneously, there is a need to adjust the levels of the individual tests to effect control of the family-wise error rate (FWER). Standard frequentist adjustments control the error rate but are typically both conservative and oblivious to prior information. We propose a Bayesian testing approach-multiplicity-calibrated Bayesian hypothesis testing-that sets individual critical values to reflect prior information while controlling the FWER via the Bonferroni inequality. If the prior information is specified correctly, in the sense that those null hypotheses considered most likely to be false in fact are false, the power of our method is substantially greater than that of standard frequentist approaches. We illustrate our method using data from a pharmacogenetic trial and a preclinical cancer study. We demonstrate its error rate control and power advantage by simulation.