Abstract: The classical Cartan-Hadamard theorem asserts that for a closed Riemannian manifold of non-positive curvature, the universal cover is always diffeomorphic to standard Euclidean space. A consequence of this is that any two such manifolds, of the same dimension, have diffeomorphic (and hence homeomorphic) universal covers. We prove an analogue of this latter statement for manifolds with boundary: given two closed negatively curved manifolds with non-empty, totally geodesic boundary, their universal covers are homeomorphic provided the manifolds have the same dimension (except possibly in the 5-dimensional situation). An application to topological rigidity of negatively curved P-manifolds will also be discussed.