In recent experiments on isogenetic cancer cell lines, it was observed that exposure to high doses of anti-cancer drugs can induce the emergence of a subpopulation of weakly-proliferative and drug-tolerant cells, that display markers associated with cancer stem cells. After a period of time, some of the surviving cells were observed to change their phenotype to resume normal proliferation, and eventually repopulate the sample. Furthermore, the drug-tolerant cells could be drug resensitized following drug washout. Here we propose a theoretical mechanism for the transient emergence of such drug tolerance. In this framework, we formulate an individual-based model and an integro-differential equation model of reversible phenotypic evolution in a cell population exposed to cytotoxic drugs. The outcomes of both models suggest that selection, non-genetic instability, stress-induced adaptation and the interplay between these mechanisms can push an actively proliferating cell population to transition into a weakly-proliferative and drug-tolerant state. Hence, the cell population experiences much less stress in the presence of the drugs and, in the long run, reacquires a proliferative phenotype, due to selection pressure and phenotypic fluctuations. These mechanisms can also reverse epigenetic drug-tolerance following drug washout. Our study highlights how the transient appearance of the weakly-proliferative and drug-tolerant cells is related to the use of high-dose therapy. Furthermore, we show how stem-like characteristics can act to stabilize the transient, weakly-proliferative and drug-tolerant subpopulation for a longer time window. Finally, using our models as in silico laboratories, we propose new testable hypotheses that could help uncover general principles underlying the emergence of cancer drug tolerance.