Fermi resonances are analysed in the literature either by degenerate first-order perturbation theory or by the coupled oscillator model of Schwartz and Wang. Here the perturbation treatment is generalized, taking into account the coupling not only between a fundamental and an overtone, but also with other eigenstates, required by a proper treatment of the Bose character of vibrations. The resulting expressions for the frequency shifts differ from those of both other models. Numerical differences from the first-order perturbation treatment are small in most cases. However, the results can be used to relate the empirical interaction constant of the coupled oscillator model to an actual anharmonicity constant.