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Systems involving nematic liquid crystals subjected to magnetic fields or electric fields are modeled using the Oseen--Frank macroscopic continuum theory, and general criteria are developed to assess the local stability of equilibrium solutions. The criteria take into account the inhomogeneity of the electric field and the mutual influence of the electric field and the liquid-crystal director field on each other. The criteria show that formulas for the instability thresholds of electric-field Fréedericksz transitions cannot in all cases be obtained from those for the analogous magnetic-field transitions by simply replacing the magnetic parameters by the corresponding electric parameters, contrary to claims in standard references. A simple analytical test is provided to determine when an electric-field-induced instability can differ qualitatively from the analogous magnetic-field-induced instability. For the systems we study, it is found that taking into account the full coupling between the electric field and the director field can either elevate or leave unchanged an instability threshold (never lower it), compared to the threshold predicted by the magnetic-field analogy. The physical mechanism that underlies the effect of elevating an instability threshold is the increase in free energy associated with a first-order change in the ground-state electric field caused by a perturbation of the ground-state director field. Examples are given that involve classical Fréedericksz transitions and also periodic instabilities. The inclusion of flexoelectric terms in the theory is studied, and it is found that these terms are not capable of altering the instability thresholds of any of the classical Fréedericksz transitions. [ABSTRACT FROM AUTHOR]