The energy liberated by fallback accretion has been suggested as a possible engine to power hydrogen-poor superluminous supernovae (SLSNe). We systematically investigate this model using the Bayesian light curve (LC) fitting code MOSFiT (Modular Open Source Fitter for Transients), fitting the LCs of 37 hydrogen-poor SLSNe assuming a fallback accretion central engine. We find that this model can yield good fits to their LCs, with a fit quality that rivals the popular magnetar engine models. Examining our derived parameters for the fallback model, we estimate the total energy requirements from the accretion disk to be 0.002-0.7 c2. If we adopt a typical conversion efficiency ∼10−3, the required mass to accrete is thus 2-700 . Many SLSNe, therefore, require an unrealistic accretion mass, and so only a fraction of these events could be powered by fallback accretion unless the true efficiency is much greater than our fiducial value. The SLSNe that require the smallest amounts of fallback mass are still fallback accretion-powered supernova candidates, but they are difficult to distinguish solely by their LC properties.