The spread of bacterial antibiotic resistance mutations is thought to be constrained by their pleiotropic fitness costs. Here we investigate the fitness costs of resistance in the context of the evolution of multiple drug resistance (MDR), by measuring the cost of acquiring streptomycin resistance mutations (StrepR) in independent strains of the bacterium Pseudomonas aeruginosa carrying different rifampicin resistance (RifR) mutations. In the absence of antibiotics, StrepR mutations are associated with similar fitness costs in different RifR genetic backgrounds. The cost of StrepR mutations is greater in a rifampicin-sensitive (RifS) background, directly demonstrating antagonistic epistasis between resistance mutations. In the presence of rifampicin, StrepR mutations have contrasting effects in different RifR backgrounds: StrepR mutations have no detectable costs in some RifR backgrounds and massive fitness costs in others. Our results clearly demonstrate the importance of epistasis and genotype-by-environment interactions for the evolution of MDR.