We examine the effects of the spontaneous background event rate and aftershock triggering characteristics on the temporal statistics of seismicity in the epidemic-type aftershock sequence model. Recent work has shown that the earthquake interevent time distribution is generally bimodal: a superposition of a gamma component from triggered aftershocks at short time intervals and an exponential component at longer intervals from spontaneous events and the overlapping of independent aftershock sequences. The relative size of these two components varies between catalogs, so there is no simple, universal scaling; at the extreme of high spontaneous rate, e.g., in large regions, the high probability of temporally overlapping aftershock sequences causes the exponential component to dominate. Here we further explore the effects of both the spontaneous rate and the aftershock triggering parameters. We show that the analytical theory of Saichev and Sornette (2007), although valid under their assumptions, gives the impression of a more "universal" behavior if used outside its stated range of applicability. We also show that within the high-overlap (high-spontaneous rate) regime, a maximum likelihood inversion of the model's temporal parameters is both less accurate and biased; specifically, the background rate is systematically overestimated. This has implications on the suitable range of region sizes for which parameter inversion may be reliable and must therefore be taken into account in any inversion for temporal variations in background rate in time-dependent hazard calculation. Copyright 2011 by the American Geophysical Union.