We consider the problem of determining whether a polytope of n??n matrices is stable, by checking stability of low-dimensional faces of the polytope. We show that stability of all (2n-4)-dimensional faces guarantees stability of the entire set. Furthermore, we prove that, for any n and any k??2n-4, there exists an unstable polytope of dimension k such that all its (2n-5)-dimensional subpolytopes are stable.