Genetic constraints can block many mutational pathways to optimal genotypes in real fitness landscapes, yet the extent to which this can limit evolution remains to be determined. Interestingly, mutator bacteria elevate only specific types of mutations, and therefore could be very sensitive to genetic constraints. Testing this possibility is not only clinically relevant, but can also inform about the general impact of genetic constraints in adaptation. Here, we evolved 576 populations of two mutator and one wild-type Escherichia coli to doubling concentrations of the antibiotic cefotaxime. All strains carried TEM-1, a b-lactamase enzyme well known by its low availability of mutational pathways. Crucially, one of the mutators does not elevate any of the relevant first-step mutations known to improve cefatoximase activity. Despite this, both mutators displayed a similar ability to evolve more than 1000-fold resistance. Initial adaptation proceeded in parallel through general multi-drug resistance mechanisms. High-level resistance, in contrast, was achieved through divergent paths; with the a priori inferior mutator exploiting alternative mutational pathways in PBP3, the target of the antibiotic. These results have implications for mutator management in clinical infections and, more generally, illustrate that limits to natural selection in real organisms are alleviated by the existence of multiple loci contributing to fitness.