The current characteristics of organic field-effect transistors (OFET) often show a disadvantageous nonlinearity at low drain voltages. It has been shown recently [J. Appl. Phys. 102, 054509 (2007)] that in top contact (TOC) OFETs this effect can be caused by trap recharging if the contacts are of Schottky type. For bottom contact (BOC) OFETs, in spite of controversial discussions, Schottky contacts as origin of the nonlinearity are often stated. At first, it is shown here by a mixed mode simulation that for large ideality factors a Schottky contact only at drain leads to such a nonlinearity. However, with the same Schottky contacts at drain and source the effect is covered by the high resistance of the contact at source. Next, the different influences of Schottky contacts on BOC OFETs and TOC OFETs with varying overlap of the source/drain contacts with the gate are clarified. Further, it is demonstrated with detailed two-dimensional simulations that the combination of the presence of Schottky contacts with a field dependence of the mobility can cause the nonlinearity. For the mobility we use the field dependent Pool/Frenkel model, and the models of Limketai et al.[Phys. Rev. B 75, 113203 (2007)] and Pasveer/Coehoorn et al.[Phys. Rev. Lett. 94, 206601 (2005)], which depend in addition on the carrier concentration. Their influence on the device performance has been clarified by the simulations. Simulated profiles of concentrations and fields lead to the understanding of the mechanism causing the nonlinearity. This mechanism is especially effective for the Pasveer/Coehoorn model. The field dependence of the mobility is a consequence of the energetic distribution of the hopping states and can hardly be avoided in solution based deposition of the active polymer layer. A strategy to prevent the nonlinearity is therefore an optimization of the contact-polymer interface such that the contacts become Ohmic.