The large diversity in MHC class I molecules in a population lowers the chance that a virus infects a host to which it is pre-adapted to escape the MHC binding of CTL epitopes. However, viruses can also lose CTL epitopes by escaping the monomorphic antigen processing components of the pathway (proteasome and TAP) that create the epitope precursors. If viruses were to accumulate escape mutations affecting these monomorphic components, they would become pre-adapted to all hosts regardless of the MHC polymorphism. To assess whether viruses exploit this apparent vulnerability, we study the evolution of HIV-1 with bioinformatic tools that allow us to predict CTL epitopes, and quantify the frequency and accumulation of antigen processing escapes. We found that within hosts, proteasome and TAP escape mutations occur frequently. However, on the population level these escapes do not accumulate: the total number of predicted epitopes and epitope precursors in HIV-1 clade B has remained relatively constant over the last 30 years. We argue that this lack of adaptation can be explained by the combined effect of the MHC polymorphism and the high specificity of individual MHC molecules. Because of these two properties, only a subset of the epitope precursors in a host are potential epitopes, and that subset differs between hosts. We estimate that upon transmission of a virus to a new host 39%-66% of the mutations that caused epitope precursor escapes are released from immune selection pressure.