Magnetic anisotropies and misfit strain relaxations have been investigated in Cu/Ni-wedge/Cu (100) and (111) sandwiches deposited by molecular beam epitaxy on single-crystal Cu substrates. Our results reveal a clear distinction in the nature of the measured anisotropy at Ni thicknesses below and above the critical value t(c), where the growth becomes incoherent. Below t(c), coherent lattice strain modifies only the volume anisotropy, while interface anisotropy is Neel type; above t(c), magnetoelastic effects are found to contribute to the interface anisotropy.