We have observed large discrepancies between the dc magnetization and ferromagnetic resonance (FMR) magnetic anisotropy in Mo-Ni superlattices. We show that higher-order anisotropy is present in both measurements and develop an analysis by which the first-order and second-order energies can be treated independently. The low-temperature results show a systematic divergence of the first-order FMR and dc anisotropy as the Ni-layer thickness decreases. It is shown that the source of the divergence is an enhanced surface anisotropy measured by FMR. Possible sources of this enhancement are discussed.