The consequence of a mutation can be influenced by the context in which it operates. For example, loss of gene function may be tolerated in one genetic background, and lethal in another. The extent to which mutant phenotypes are malleable, the architecture of modifiers and the identities of causal genes remain largely unknown. Here, we measure the fitness effects of ~ 1,100 temperature-sensitive alleles of yeast essential genes in the context of variation from ten different natural genetic backgrounds and map the modifiers for 19 combinations. Altogether, fitness defects for 149 of the 580 tested genes (26%) could be suppressed by genetic variation in at least one yeast strain. Suppression was generally driven by gain-of-function of a single, strong modifier gene, and involved both genes encoding complex or pathway partners suppressing specific temperature-sensitive alleles, as well as general modifiers altering the effect of many alleles. The emerging frequency of suppression and range of possible mechanisms suggest that a substantial fraction of monogenic diseases could be managed by modulating other gene products.