The accumulation of mutations in RNA viruses is thought to facilitate rapid adaptation to changes in the environment. However, most mutations have deleterious effects on fitness, especially for viruses. Thus, tolerance to mutations should determine the nature and extent of genetic diversity that can be maintained in the population. Here, we combine population genetics theory, computer simulation, and experimental evolution to examine the advantages and disadvantages of tolerance to mutations, also known as mutational robustness. We find that mutational robustness increases neutral diversity and, as expected, can facilitate adaptation to a new environment. Surprisingly, under certain conditions, robustness may also be an impediment for viral adaptation, if a highly diverse population contains a large proportion of previously neutral mutations that are deleterious in the new environment. These findings may inform therapeutic strategies that cause extinction of otherwise robust viral populations.