Mixed quantum-classical methods are applied to an increasingly challenging series of model problems, and their accuracy is examined. The models involve one light and one heavy degree of freedom, and exhibit substantial nonadiabatic behavior. In all of the models the coupling between the light and heavy particles is linear (harmonic), In addition, different external potentials are applied to the heavy particle only. The energies of the light particle quantum states, as a function of the position of the heavy particle, define a sequence of ground and excited Born-Oppenheimer potential energy curves. Because the light particle experiences a purely harmonic potential, the potential energy curves are parallel and equally spaced for all of the models. In addition, the nonadiabatic couplings among potential energy curves persist for all times due to the nonvanishing linear coupling between light and heavy particles. The model problems were used to test two strategies for carrying out mixed quantum-classical dynamics in systems involving nonadiabatic transitions: mean field and surface hopping. The model calculations reported here suggest that, in cases where linear couplings dominate, the mean field mixed quantum-classical method displays useful accuracy and is robust to the introduction of anharmonic heavy-particle interactions. The model calculations also reveal special situations in which the surface hopping approximation is inappropriate.