The question of what drives tectonic plates has been revitalized by seismic observations that cannot be explained by conventional plate-driving forces. The observations, designed to constrain flow in the asthenosphere, are consistent with the asthenosphere locally flowing faster than the plate above and in a direction offset from plate motion. These inferences are not consistent with plates being driven exclusively by slab-pull and/or ridge-push forces. Mantle convection models were put forth to argue that pressure-driven flow, interacting with a non-Newtonian upper mantle viscosity, could explain these observations. To test the robustness of those results, we expand the models to allow for the development of weak plate margins and associated plate-like behavior. We find that with weak margins, the overall component of slab-driven flow becomes stronger while pressure driven asthenosphere flow remains active. Locally, the asthenosphere can lead plates and there are rotations in the direction of asthenosphere flow with depth. The balance of plate driving forces (i.e., the ratio of slab-pull to asthenosphere flow) is found to depend on plate margin strength. The models also indicate that a non-Newtonian upper mantle allows for a hysteresis effect such that, depending on initial conditions, single-plate and plate-tectonic modes can exist at the same parameter conditions.