Overexpression of a drug's molecular target often increases drug resistance, offering a pathway for adaptive evolution and a tool for target identification. It is unclear though why this phenomenon applies to some drugs but not others. Here we gradually overexpressed antibiotic targets in Escherichia coli and found that drug resistance can increase, remain unchanged, decrease or even change non-monotonically. Even a single target can produce opposing responses to its different inhibitors. We explain these contradicting effects with quantitative models of enzyme inhibition that account for fitness costs and the biochemical activity or inactivity of drug-enzyme complexes. Thus, target overexpression confers resistance or sensitivity as a predictable property of drug mechanism, explaining its variable presence in nature as a resistance mechanism. Though overexpression screens may fail at identifying unknown targets, overexpressing known or putative targets provides a systematic approach to distinguish between simple inhibition and complex mechanisms of drug action.