Recent work indicates that the strong cosmic censorship hypothesis is violated by nearly extremal Reissner-Nordstrom-de Sitter black holes. It was argued that perturbations of such a black hole decay sufficiently rapidly that the perturbed spacetime can be extended across the Cauchy horizon as a weak solution of the equations of motion. In this paper we consider the case of Kerr-de Sitter black holes. We find that, for any non-extremal value of the black hole parameters, there are quasinormal modes which decay sufficiently slowly to ensure that strong cosmic censorship is respected. Our analysis covers both scalar field and linearized gravitational perturbations.