We consider a lattice model whose spins may assume a finite number q of values. The interaction energy between two nearest-neighbor spins takes on the value J_1 + J_2 or J_2, depending on whether the two spins coincide or are different but coincide modulo q_1, and it is zero otherwise. This model is a generalization of the Ashkin-Teller model and exhibits the multilayer wetting phenomenon, that is, wetting by one or two or three interfacial layers, depending on the number of phases in coexistence.