The evolution of drug resistance is a major health threat. In chronic infections with rapidly mutating pathogens—including HIV, tuberculosis, and hepatitis B and C viruses—multidrug resistance can cause even aggressive combination drug treatment to fail. Oftentimes, individual drugs within a combination do not penetrate equally to all infected regions of the body. Here we present a mathematical model suggesting that this imperfect penetration can dramatically increase the chance of treatment failure by creating regions where only one drug from a combination reaches a therapeutic concentration. The resulting single-drug compartments allow the pathogen to evolve resistance to each drug sequentially, rapidly causing multidrug resistance. More broadly, our model provides a quantitative framework for reasoning about trade-offs between aggressive and moderate drug therapies.