Cytotoxic T cells (CTLs) perceive the world through small peptides that are eight to ten amino acids long. These peptides (epitopes) are initially generated by the proteasome, a multi-subunit protease that is responsible for the majority of intra-cellular protein degradation. The proteasome generates the exact C-terminal of CTL epitopes, and the N-terminal with a possible extension. CTL responses may diminish if the epitopes are destroyed by the proteasomes. Therefore, the prediction of the proteasome cleavage sites is important to identify potential immunogenic regions in the proteomes of pathogenic microorganisms (or humans). We have recently shown that NetChop, a neural network-based prediction method, is the best method available at the moment to do such predictions; however, its performance is still lower than desired. Here, we use novel sequence encoding methods and show that the new version of NetChop predicts approximately 10% more of the cleavage sites correctly while lowering the number of false positives with close to 15%. With this more reliable prediction tool, we study two important questions concerning the function of the proteasome. First, we estimate the N-terminal extension of epitopes after proteasomal cleavage and find that the average extension is relatively short. However, more than 30% of the peptides have N-terminal extensions of three amino acids or more, and thus, N-terminal trimming might play an important role in the presentation of a substantial fraction of the epitopes. Second, we show that good TAP ligands have an increased chance of being cleaved by the proteasome, i.e., the specificity of TAP has evolved to fit the specificity of the proteasome. This evolutionary relationship allows for a more efficient antigen presentation.