We construct a model to study tradeoffs associated with aging in the adaptive immune system, focusing on cumulative effects of replacing naive cells with memory cells. Binding affinities are characterized by a stochastic shape space model. System loss arising from an individual infection is associated with disease severity, as measured by the total antigen population over the course of an infection. We monitor evolution of cell populations on the shape space over a string of infections, and find that the distribution of losses becomes increasingly heavy-tailed with time. Initially this lowers the average loss: the memory cell population becomes tuned to the history of past exposures, reducing the loss of the system when subjected to a second, similar infection. This is accompanied by a corresponding increase in vulnerability to novel infections, which ultimately causes the expected loss to increase due to overspecialization, leading to increasing fragility with age (i.e., immunosenescence). In our model, immunosenescence is not the result of a performance degradation of some specific lymphocyte, but rather a natural consequence of the built-in mechanisms for system adaptation. This robust, yet fragile behavior is a key signature of Highly Optimized Tolerance.