Charge transport in disordered organic semiconductors, which is governed by incoherent hopping between localized molecular states, is frequently studied using a mean-field approach. However, such an approach only considers the time-averaged occupation of sites and neglects the correlation effect resulting from the Coulomb interaction between charge carriers. Here, we study the charge transport in unipolar organic devices using kinetic Monte Carlo simulations and show that the effect of Coulomb correlation is already important when the charge-carrier concentration is above ${10}^{-3}$ per molecular site and the electric field is smaller than ${10}^8$ V/m. The mean-field approach is then no longer valid, and neglecting the effect can result in significant errors in device modeling. This finding is supported by experimental current density-voltage characteristics of ultrathin sandwich-type unipolar poly(3-hexylthiophene) (P3HT) devices, where high carrier concentrations are reached.