Prokaryotic CRISPR-Cas adaptive immune systems insert spacers derived from viruses and other parasitic DNA elements into CRISPR loci to provide sequence-specific immunity1,2. This frequently results in high within-population spacer diversity3–6, but it is unclear if and why this is important. Here we show that, as a result of this spacer diversity, viruses can no longer evolve to overcome CRISPR-Cas by point mutation, which results in rapid virus extinction. This effect arises from synergy between spacer diversity and the high specificity of infection, which greatly increases overall population resistance. We propose that the resulting short-lived nature of CRISPR-dependent bacteria–virus coevolution has provided strong selection for the evolution of sophisticated virus-encoded anti-CRISPR mechanisms7.