Understanding adaptation by natural selection requires understanding the genetic factors that determine which beneficial mutations are available for selection. Here, using experimental evolution of rifampicin-resistant Pseudomonas aeruginosa, we show that different genotypes vary in their capacity for adaptation to the cost of antibiotic resistance. We then use sequence data to show that the beneficial mutations associated with fitness recovery were specific to particular genetic backgrounds, suggesting that genotypes had access to different sets of beneficial mutations. When we manipulated the supply rate of beneficial mutations, by altering effective population size during evolution, we found that it constrained adaptation in some selection lines by restricting access to rare beneficial mutations, but that the effect varied among the genotypes in our experiment. These results suggest that mutational neighbourhood varies even among genotypes that differ by a single amino acid change, and this determines their capacity for adaptation as well as the influence of population biology processes that alter mutation supply rate.