We investigate the conditions under which the stress-release model, a stochastic version of the elastic rebound model, produces synthetic earthquake sequences characterized by Accelerating Seismic Release (ASR). In this model, the level, or stress, of the process accumulates linearly with time through tectonic input and decreases as the result of earthquakes. These stress drops correspond to some power of the energy released in the earthquakes, either E0.5 (Benioff strain) or E (seismic moment). Earthquakes occur in a point process with rate controlled by the level of the process. We hypothesize that the critical factor in the appearance of ASR is the manner in which the event sizes depend on the level of the process. This is modeled by the square root of energy released following either a tapered Pareto or truncated Gutenberg-Richter distribution, with maximum earthquake size controlled by a tail-off or truncation point. As the tail-off point becomes large, so does the average size, corresponding to an acceleration to criticality of the system. We found that those cases where the underlying level of the process corresponded to accumulated seismic moment produced numerous ASR sequences, whereas those cases using accumulated Benioff strain as the level did not. These results suggest that the occurrence of ASR is strongly dependent on how large earthquakes affect the dynamics of the fault system in which they are embedded, and hopefully provide some insight into the mechanics of acceleration to criticality, i.e., on the possible causes of occurrence/nonoccurrence of ASR.