The amplitude dependence of the relative AC losses q in anisotropic superconducting plate-like samples has been studied experimentally and theoretically. Contrary to the isotropic case where there exists one maximum in the dependence of q=χ′′ on the amplitude h0 of the AC magnetic field, two maxima in the dependence q(h0) were observed. Both maxima are found to be related to the penetration of the AC field up to the middle of a sample. This size effect occurs if the direction of the AC magnetic field differs significantly from the principal axes of symmetry. Such an unusual manifestation of the size effect can be understood under the assumption that two components of the AC field along the principal axes of symmetry are independently shielded by the two principal components of the anisotropic critical current density. The obtained results are interpreted within the framework of the critical state model generalized to the anisotropic case.