In order to recognize and combat a diverse array of pathogens the immune system has a large repertoire of T cells having unique T cell receptors (TCRs) with only a few clones specific for any given antigen. We discuss how the number of different possible TCRs encoded in the genome (the potential repertoire) and the number of different TCRs present in an individual (the realized repertoire) can be measured. One puzzle is that the potential repertoire greatly exceeds the realized diversity of naïve T cells within any individual. We show that the existing hypotheses fail to explain why the immune system has the potential to generate far more diversity than is used in an individual, and propose an alternative hypothesis of “evolutionary sloppiness.” Another immunological puzzle is why mice and humans have similar repertoires even though humans have over 1000-fold more T cells. We discuss how the idea of the “protecton,” the smallest unit of protection, might explain this discrepancy and estimate the size of “protecton” based on available precursor frequencies data. We then consider T cell cross-reactivity – the ability of a T cell clone to respond to more than one epitope. We extend existing calculations to estimate the extent of expected cross-reactivity between the responses to different pathogens. Our results are consistent with two observations: a low probability of observing cross-reactivity between the immune responses to two randomly chosen pathogens; and the ensemble of memory cells being sufficiently diverse to generate cross-reactive responses to new pathogens.