Populations need to adapt to sustained climate change, which requires micro-evolutionary change in the long term. A key question is how the rate of this micro-evolutionary change compares with the rate of environmental change, given that theoretically there is a ‘critical rate of environmental change’ beyond which increased maladaptation leads to population extinction. Here, we parametrize two closely related models to predict this critical rate using data from a long-term study of great tits (Parus major). We used stochastic dynamic programming to predict changes in optimal breeding time under three different climate scenarios. Using these results we parametrized two theoretical models to predict critical rates. Results from both models agreed qualitatively in that even ‘mild’ rates of climate change would be close to these critical rates with respect to great tit breeding time, while for scenarios close to the upper limit of IPCC climate projections the calculated critical rates would be clearly exceeded with possible consequences for population persistence. We therefore tentatively conclude that micro-evolution, together with plasticity, would rescue only the population from mild rates of climate change, although the models make many simplifying assumptions that remain to be tested.