Drug tolerant persisters (DTPs) drive cancer therapy resistance by temporarily evading drug action, allowing multiple routes to eventual permanent resistance. Despite clear evidence for DTPs, the timing of their emergence, proliferative nature, and how their population dynamics arise from measured single-cell kinetics remain poorly understood. Here we use time-lapse microscopy data from two cancer cell lines, integrating single-cell and population measurements, to develop a quantitative description of drug persistence. Contrary to the expectation that increasing levels of genotoxic stress should lead to slower times to division and faster times to death, we observe minor changes in the single-cell intermitotic and death time distributions upon increasing cisplatin concentration. Yet, population decay rates increase 3-fold, suggesting a surprising independence of the overall dynamics from the measured birth and death rates. To explain this phenomenon, we argue that the observed lineage correlations and concentration-dependent decay rates imply cell-state dependent fate choices made both pre and post-cisplatin as opposed to just post-drug birth/death rate-based competitive fate choices. We demonstrate that these cell-states, present in the pre-drug ancestors of DTP and sensitive cells, exhibit no difference in cycling speed and are inherited across at least 2-3 cellular generations. Post-drug survival versus death fates are decided with high probability by these pre-existing cell-states, but get modulated to some extent by the drug, leading to a drug concentration dependent state-fate map. A stochastic model implementing these rules simultaneously recapitulates the observed decay rates and cell-fate lineage correlations. The model also demonstrates how the use of barcode diversity change before and after drug might lead to misleading interpretations of the timing of persister fate decisions. Our results provide a conceptual framework for quantifying pre versus post-drug contributions to cell fate, without requiring knowledge of the underlying molecular architecture of the heterogeneous cell states.