Massive bosonic fields of arbitrary spin are predicted by general extensions of the standard model. It has been recently shown that there exists a family of bimetric theories of gravity-including massive gravity-which are free of Boulware-Deser ghosts at the nonlinear level. This opens up the possibility to describe consistently the dynamics of massive spin-2 particles in a gravitational field. Within this context, we develop the study of massive spin-2 fluctuations-including massive gravitons-around Schwarzschild and slowly rotating Kerr black holes. Our work has two important outcomes. First, we show that the Schwarzschild geometry is linearly unstable for small tensor masses, against a spherically symmetric mode. Second, we provide solid evidence that the Kerr geometry is also generically unstable, both against the spherical mode and against long-lived superradiant modes. In the absence of nonlinear effects, the observation of spinning black holes bounds the graviton mass mu to be mu less than or similar to 5 x 10(-23) eV.