The last few years have seen numerous record-shattering heatwaves that seem to occur in all corners of the globe. In the aftermath of these devastating and damaging events, there is considerable interest in identifying worst-case thresholds or upper bounds that quantify just how hot temperatures may become. Atmospheric theory provides physical upper bounds on extreme temperatures, but these generally correspond to idealized conditions that may rarely occur in reality. Generalized extreme value (GEV) theory provides a data-driven estimate of thresholds for extremes: when the GEV shape parameter is negative, the distribution has a finite upper bound. In either case it is possible for historical estimates of this upper bound to be exceeded by future events, which precludes robust attribution statements and undermines planning for the impacts of extreme heatwaves. Here, we show how the frequency and relative extremity of observed events globally that exceed a GEV upper bound based on prior events, so-called “impossible” heatwaves, have changed over time. We find clear connections between the frequency of “impossible” events and anthropogenic forcing, such that their occurrence is more probable now versus a pre-industrial climate. However, we also find that many events that have been termed “impossible” are actually within data-driven upper bound estimates, but only when using scalable statistical and machine learning methods that account for physical covariates and the spatial coherence of individual heatwave events. Robust understanding of these thresholds and occurrence probabilities provides critical information about record-breaking heatwave events that have not yet occurred and how their magnitude relates to historical measurements.