We address the question of whether a graviton could have a small nonzero mass. The issue is subtle for two reasons: there is a discontinuity in the mass in the lowest tree-level approximation, and, moreover, the non-linear four-dimensional theory of a massive graviton is not defined unambiguously. First, we reiterate the old argument that for vanishing graviton mass the lowest tree-level approximation breaks down since the higher order corrections are singular in the graviton mass. However, there can exist nonperturbative solutions which correspond to the summation of the singular terms, and these solutions are continuous in the graviton mass. Furthermore, we study a completely nonlinear and generally covariant five-dimensional model which mimics the properties of the four-dimensional theory of massive gravity. We show that the exact solutions of the model are continuous in the mass, yet the perturbative expansion exhibits a discontinuity in the leading order and singularities in higher orders as in the four-dimensional case. Based on exact cosmological solutions of the model we argue that the helicity-zero graviton state responsible for the perturbative discontinuity decouples from the matter in the limit of vanishing graviton mass in the full classical theory.