The pacing of glacial-interglacial cycles during the Quaternary period (the past 2.6 million years) is attributed to astronomically driven changes in high-latitude insolation. However, it has not been clear how astronomical forcing translates into the observed sequence of interglacials. Here we show that before one million years ago interglacials occurred when the energy related to summer insolation exceeded a simple threshold, about every 41,000 years. Over the past one million years, fewer of these insolation peaks resulted in deglaciation (that is, more insolation peaks were 'skipped'), implying that the energy threshold for deglaciation had risen, which led to longer glacials. However, as a glacial lengthens, the energy needed for deglaciation decreases. A statistical model that combines these observations correctly predicts every complete deglaciation of the past million years and shows that the sequence of interglacials that has occurred is one of a small set of possibilities. The model accounts for the dominance of obliquity-paced glacial-interglacial cycles early in the Quaternary and for the change in their frequency about one million years ago. We propose that the appearance of larger ice sheets over the past million years was a consequence of an increase in the deglaciation threshold and in the number of skipped insolation peaks.