Recent experiments have suggested that high energy charge carriers can have a significant effect on the operation of group III nitride (III-N) light-emitting diodes (LEDs), possibly playing an important role in the efficiency droop. As hot carriers are not accounted for by device simulation tools based on drift-diffusion (DD) and quasiequilibrium conditions, more advanced tools are needed. Here, fully self-consistent Monte Carlo (MC) simulations are developed to investigate the effects of hot carriers in device operation and to outline the shortcomings of the DD models in modeling multiquantum well (MQW) LEDs. The results show that hot carrier transport can lead to substantial electron overflow distributing the carriers more evenly in MQW structures, increasing the total recombination and leakage currents. Also Auger recombination is found to drive the distributions out of quasiequilibrium but, surprisingly, it does not contribute extensively to the leakage current. The simulations involve in-house ab-initio band structures as well as parameterized band structures, but qualitatively the results do not strongly depend on the band structure details. However, there is a clear discrepancy between the DD and MC simulations at bias voltages significantly exceeding the built-in potential when the LED consists of several deep quantum wells.