Measurements on protein translocation through solid-state nanopores reveal anomalous (non-Smoluchowski) transport behavior, as evidenced by extremely low detected event rates; that is, the capture rates are orders of magnitude smaller than what is theoretically expected. Systematic experimental measurements of the event rate dependence on the diffusion constant are performed by translocating proteins ranging in size from 6 to 660 kDa. The discrepancy is observed to be significantly larger for smaller proteins, which move faster and have a lower signal-to-noise ratio. This is further confirmed by measuring the event rate dependence on the pore size and concentration for a large 540 kDa protein and a small 37 kDa protein, where only the large protein follows the expected behavior. We dismiss various possible causes for this phenomenon and conclude that it is due to a combination of the limited temporal resolution and low signal-to-noise ratio. A one-dimensional first-passage time-distribution model supports this and suggests that the bulk of the proteins translocate on time scales faster than can be detected. We discuss the implications for protein characterization using solid-state nanopores and highlight several possible routes to address this problem.