In this study I show that simple heuristic models and numerical calculations suggest that an entire class of commonly invoked models of earthquake failure processes cannot explain triggering of seismicity by transient or "dynamic" stress changes, such as stress changes associated with passing seismic waves. The models of this class have the common feature that the physical property characterizing failure increases at an accelerating rate when a fault is loaded (stressed) at a constant rate. Examples include models that invoke rate state friction or subcritical crack growth, in which the properties characterizing failure are slip or crack length, respectively. Failure occurs when the rate at which these grow accelerates to values exceeding some critical threshold. These accelerating failure models do not predict the finite durations of dynamically triggered earthquake sequences (e.g., at aftershock or remote distances). Some of the failure models belonging to this class have been used to explain static stress triggering of aftershocks. This may imply that the physical processes underlying dynamic triggering differs or that currently applied models of static triggering require modification. If the former is the case, we might appeal to physical mechanisms relying on oscillatory deformations such as compaction of saturated fault gouge leading to pore pressure increase, or cyclic fatigue. However, if dynamic and static triggering mechanisms differ, one still needs to ask why static triggering models that neglect these dynamic mechanisms appear to explain many observations. If the static and dynamic triggering mechanisms are the same, perhaps assumptions about accelerating failure and/or that triggering advances the failure times of a population of inevitable earthquakes are incorrect.