We prove the existence of multiple bound states of the nonlinear Schrödinger equation −∆u + V (x)u = f (u). Here the linear potential V is continuous and bounded from below, and the nonlinearity f is of asymptotically linear type. We show that, under certain assumptions on the spectrum of the Schrödinger operator −∆ + V and the asymptotic behaviour of f (u) u , the above equation has at least four nontrivial solutions, two of them sign changing.