The relationship between biological research and mathematical modeling is complex, critical, and vital. In this review, we summarize the results of the collaboration between two laboratories, exploring the interaction between mathematical modeling and wet-lab immunology. During this collaboration several aspects of the immune defence against viral infections were investigated, focusing primarily on the subject of heterologous immunity. In this manuscript, we emphasize the topics where computational simulations were applied in conjunction with experiments, such as immune attrition, the growing and shrinking of cross-reactive T cell repertoires following repeated infections, the short and long-term effects of cross-reactive immunological memory, and the factors influencing the appearance of new clonal specificities. For each topic, we describe how the mathematical model used was adapted to answer specific biological questions, and we discuss the hypotheses that were generated by simulations. Finally, we propose rules for testing hypotheses that emerge from model experimentation in the wet lab, and vice-versa.