Analysis of tipping points (rapid, large-scale, and potentially irreversible transitions in Earth system components) is crucial for assessing the resilience of the Earth’s subsystems and anticipating risks associated with climate change. Predicting tipping points from time-series data requires accounting for variability, system-specific factors, and high-quality data, which are often limited. Existing approaches, while useful, depend on strong assumptions that may reduce predictive accuracy. In this study, we focus on the model used to approximate system behavior, and argue that modeling assumptions can significantly alter estimates of critical thresholds, and including assessments of whether a system will reach a critical transition at all. We show that, in the case of the Atlantic Meridional Overturning Circulation (AMOC), assuming different polynomial degrees in simple model approximations can lead to entirely different estimates of the critical threshold, or even that no threshold exists. This effect is particularly exacerbated by the interannual variability of the AMOC fingerprints used. These considerations highlight the need for more advanced techniques and increased robustness in model selection and underlying assumptions when interpreting estimates of critical transitions.