There is a theorem from Cheeger-Colding saying the following:
Let $n$ be an integer. If a sequence of $n$-dimensional Riemannian manifolds $(M_i,g_i)$ converges with respect to the Gromov-Hausdorff to an $n$-dimensional compact riemannian manifold $(M,g)$, with a uniform lower bound on the Ricci curvature, that is: $ Ric(M_i) \geq -(n-1) g_i $ then there is a rank $i_0$ such that for all $i \geq i_0$, $M_i$ is diffeomorphic to $M$.
So my question is: Is somebody aware of a situation where a sequence of manifolds satisfiying the hypothesis DOES NOT converges to $M$ with respect to the Lipschitz distance as well ? (that is, the diffeomorphisms given by the theorem cannot be supposed to be arbitrarily close to isometries as the rank increases)
All I know is that M. Anderson has shown that when Ricci curvature is also uniformly bounded above then Lipschitz convergence does hold.