The time dependence of heating in solar active regions can be studied by analyzing the slope of the emission measure distribution coolward of the peak. In a previous study we showed that low-frequency heating can account for 0% to 77% of active region core emission measures.We now turn our attention to heating by a finite succession of impulsive events for which the timescale between events on a single magnetic strand is shorter than the cooling timescale.We refer to this scenario as a “nanoflare train” and explore a parameter space of heating and coronal loop properties with a hydrodynamic model. Our conclusions are (1) nanoflare trains are consistent with 86% to 100% of observed active region cores when uncertainties in the atomic data are properly accounted for; (2) steeper slopes are found for larger values of the ratio of the train duration ΔH to the post-train cooling and draining timescale ΔC, where ΔH depends on the number of heating events, the event duration and the time interval between successive events (τC); (3) τC may be diagnosed from the width of the hot component of the emission measure provided that the temperature bins are much smaller than 0.1 dex; (4) the slope of the emission measure alone is not sufficient to provide information about any timescale associated with heating—the length and density of the heated structure must be measured for ΔH to be uniquely extracted from the ratio ΔH /ΔC.