Increasing ice flux from glaciers retreating over deepening bed topography has been implicated in the recent acceleration of mass loss from the Greenland and Antarctic ice sheets. We show in observations that some glaciers have remained at peaks in bed topography without retreating despite enduring significant changes in climate. Observations also indicate that some glaciers which persist at bed peaks undergo sudden retreat years or decades after the onset of local ocean or atmospheric warming. Using model simulations, we show that glacier persistence may lead to two very different futures: one where glaciers persist at bed peaks indefinitely, and another where glaciers retreat from the bed peak suddenly without a concurrent climate forcing. However, it is difficult to distinguish which of these two futures will occur from current observations. We conclude that inferring glacier stability from observations of persistence obscures our true commitment to future sea-level rise under climate change.