Consider the initial-boundary value problem for the 2-speed Carleman model of the Boltzmann equation of the kinetic theory of gases set in some bounded interval with boundary conditions prescribing the density of particles entering the interval. Under the usual parabolic scaling, a nonlinear diffusion limit is established for this problem. In fact, the techniques presented here allow treating generalizations of the Carleman system where the collision frequency is proportional to some power of the macroscopic density, with exponent in [-1,1].