Hyperbolic manifolds have constant sectional curvature $-1$. The two-holed torus, for example, can (I believe) be given a hyperbolic metric so that it has curvature $-1$. It should also be a complete metric space, with this metric.
It would seem as though this should make the two-holed torus with this metric a CAT(0) space (even a CAT(-1) space). However, the two-holed torus is obviously not contractible, but CAT(0) spaces are contractible.
I'm sure I'm just confused about the definitions, but I'm not sure where.
These manifolds are locally CAT(-1) (and, hence, locally CAT(0)). Note that locally CAT(k) spaces are also said to have curvature $\le k$. To make them globally CAT(0) you need to add "complete and simply connected". ("Complete" will be automatic if your manifold is compact.)