Using long, all-atom simulations enabled by special-purpose hardware, we studied K+ permeation at the level of individual permeation events across the bacterial channel gramicidin A and across the KV1.2/2.1 voltage-gated potassium channel. At experimentally accessible voltages, which include the physiological range, the simulated permeation rates were substantially lower than the experimentally observed rates for both systems. In addition, the current-voltage relationships were nonlinear, but became linear at much higher, non-physiological voltages. In gramicidin A, the underestimated permeation rate largely resulted from too-infrequent ion recruitment into the pore lumen, although reducing the interaction strength between the ion and the pore did increase the observed permeation rate. In KV1.2/2.1, we observed at all voltages permeation consistent with a knock-on mechanism, and found that the predicted rate was lower than the experimental rate because the knock-on intermediate formed too infrequently. Additional simulations further suggested that the knock-on permeation mechanism in different K+ channels could vary, possibly due to sequence and structural variations in the selectivity-filter regions of these channels. Despite the need to apply large voltages to simulate the permeation processes, the apparent voltage insensitivity of the permeation mechanisms overall suggests that the direct simulation of permeation at the single-ion level can provide fundamental physiological insight into ion channel function. Polarizable force fields and membrane models with improved dipolar potential and dielectric constants would be needed, however, to obtain accurate simulated permeation rates at experimentally accessible voltages.