We investigate strong coupling effects in a covariant massive gravity model, which is a candidate for a ghost-free nonlinear completion of Fierz-Pauli. We analyze the conditions to recover general relativity via the Vainshtein mechanism in the weak field limit, and find three main cases depending on the choice of parameters. In the first case, the potential is such that all nonlinearities disappear and the vDVZ discontinuity cannot be avoided. In the second case, the Vainshtein mechanism allows to recover general relativity within a macroscopic radius from a source. In the last case, the strong coupling of the scalar graviton completely shields the massless graviton, and weakens gravity when approaching the source. In the second part of the paper, we explore new exact vacuum solutions, that asymptote to de Sitter or anti de Sitter space depending on the choice of parameters. The curvature of the space is proportional to the mass of the graviton, thus providing a cosmological background which may explain the present-day acceleration in terms of the graviton mass. Moreover, by expressing the potential for nonlinear massive gravity in a convenient form, we also suggest possible connections with a higher-dimensional framework.