We consider a family of degenerate elliptic equations of the form div(del F(del u)) = f, where F is an element of C-1,C-1 is a convex function which is elliptic outside a ball. We prove an excess-decay estimate at points where del u is close to a nondegenerate value for F. This result applies to degenerate equations arising in traffic congestion, where we obtain continuity of del u outside the degeneracy, and to anisotropic versions of the p-laplacian, where we get Holder regularity of del u.