The scaling of rupture properties with magnitude is of critical importance to earthquake early warning (EEW) systems that rely on source characterization using limited snapshots of waveform data. ShakeAlert, a prototype EEW system that is being developed for the western United States, provides real-time estimates of earthquake magnitude based on P-wave peak ground displacements measured at stations triggered by the event. The algorithms used in ShakeAlert assume that the displacement measurements at each station are statistically independent and that there exists a linear and time-independent relation between log peak ground displacement and earthquake magnitude. Here we challenge this basic assumption using a comprehensive database of more than 130,000 vertical component waveforms from M4.5-M9 earthquakes occurring near Japan from 1997 through 2017 and recorded by the K-NET and KiK-net strong-motion networks. By analyzing the time-evolution of P-wave peak ground displacements for these earthquakes, we show that there is a break, or saturation, in the magnitude-displacement scaling that depends on the length of the measurement time window. We demonstrate that the magnitude at which this saturation occurs is well-explained by a simple and non-deterministic model of earthquake rupture growth. We then use the predictions of this saturation model to develop a Bayesian framework for estimating posterior uncertainties in real-time magnitude estimates which incorporates the expected time-dependence of the peak displacement measurements.