Critical temperatures are widely used to quantify the upper and lower thermal limits of organisms. But measured critical temperatures often vary with methodological details, leading to spirited discussions about the potential consequences of stress and acclimation during the experiments. We review a model based on the simple assumption that failure rate increases with increasing temperature, independent of previous temperature exposure, water loss or metabolism during the experiment. The model predicts that mean critical thermal maximal temperatures (CTmax) increases nonlinearly with starting temperature and ramping rate, a pattern frequently observed in empirical studies. We then develop a statistical model that estimates a failure rate function (the relationship between failure rate and current temperature) using maximum likelihood; the best model accounts for 58% of the variation in CTmax in an exemplary dataset for tsetse flies. We then extend the model to incorporate potential effects of stress and acclimation on the failure rate function; the results show how stress accumulation at low ramping rate may increase the failure rate and reduce observed values of CTmax. We also applied the model to an acclimation experiment with hornworm larvae that used a single starting temperature and ramping rate; the analyses show that increasing acclimation temperature significantly reduced the slope of the failure rate function, increasing the temperature at which failure occurred. The model directly applies to critical thermal minima, and can utilize data from both ramping and constant temperature assays. Our model provides a new approach to analyzing and interpreting critical temperatures.