  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 to be
smaller than 5x10^{-23} eV.
