Using continuum and statistical mechanical theories, we study the switching properties of a ferronematic in a nematic liquid crystal cell subject to homeotropic boundary conditions at the cell and particle walls. An external magnetic field normal to the cell plane is also imposed. At low fields we find thresholdless switching of the nematic director, consistent with experimental data. At higher fields, there are three regimes, depending on the strength of the anchoring interaction between the director and the ferroparticle orientation. For low anchoring strengths, there is an inverse Frederiks effect, and the nematic reorientation reduces and then disappears continuously at a critical magnetic field. At intermediate fields, the degree of reorientation reduces at high fields but remains finite. For high fields, however, the director switching saturates. The dimensionless temperature scale in the problem involves the temperature, the mean nematic elastic constant, the colloidal density, and the cell dimension. If this quantity is sufficiently low, then high magnetic fields can cause magnetic segregation. The segregation order parameter is coupled to the director distortion, and this can change the inverse Frederiks transition into a first order transition, leading to bistability in an intermediate field regime. These features are perturbed but not changed structurally by the effect of a small bias magnetic field (< 10 Oe) normal to the unperturbed director. Subject to suitable choice of parameters, the theory is also quantitatively consistent with the results of the classic experiment of Chen and Amer in 1983.