Mathematical models are used to examine the relationship between checkpoint competence, ageing, and the development of cancer. The models take into account the dynamics of healthy tissue, the dynamics of initial tumor growth, and the interactions between healthy tissue and tumor cells. Two types of behavior are found. (i) A reduction of checkpoint competence results in reduced ageing of tissue, but in faster development and progression of tumors. (ii) Reduced checkpoint competence results both in reduced ageing of tissue, and in a reduced incidence of tumors. The tumors which do become established, however, are predicted to progress at an accelerated rate. The models define the conditions under which this counter-intuitive finding is observed. One reason could be the relationship between checkpoint activity and the ability of the tissue environment to exert inhibitory effects on tumor cells. Checkpoints induce senescence in tissue cells, and this compromises their ability to suppress tumor growth. Reduced checkpoint competence leads to reduced tissue senescence, and this results in higher amounts of tumor inhibition. The theoretical work is discussed with respect to data from p53 mutant mice, which show both types of relationships suggested by the models. The models help to identify differences in the experimental protocols which could explain the seemingly contradictory observations.