It is generally accepted that some screen-detected breast cancers are overdiagnosed and would not progress to symptomatic cancer if left untreated. However, precise estimates of the fraction of nonprogressive cancers remain elusive. In recognition of the weaknesses of overdiagnosis estimation methods based on excess incidence, there is a need for model-based approaches that accommodate nonprogressive lesions. Here, we present an in-depth analysis of a generalized model of breast cancer natural history that allows for a mixture of progressive and indolent lesions. We provide a formal proof of global structural identifiability of the model and use simulation to identify conditions that allow for parameter estimates that are sufficiently precise and practically actionable. We show that clinical follow-up after the last screening can play a critical role in ensuring adequately precise identification of the fraction of indolent cancers in a stop-screen trial design, and we demonstrate that model misspecification can lead to substantially biased estimates of mean sojourn time. Finally, we illustrate our findings using the example of Canadian National Breast Screening Study 2 (1980-1985) and show that the fraction of indolent cancers is not precisely identifiable. Our findings provide the foundation for extended models that account for both in situ and invasive lesions.