Gromov compactness theorem states that in a class of Riemannian manifolds that have a uniformly bounded diameter and uniformly bounded below Ricci curvature every sequence of manifolds has a subsequence that has a limit in the Gromov-Hausdorff metric.
What are the counterexamples to this statement if I drop the boundedness of Ricci curvature? That is, I am looking for a sequence of Riemannian manifolds with arbitrary negative Ricci curvature of bounded diameter and such that no subsequence is converging in GH sense.
Without the Ricci curvature bound one does not have control on the doubling constant of the space (which is the only thing that the bound is used for). This means that the spaces may contain larger and larger sets of uniformly separated points (say, distance $\ge 1$ between any two points). This precludes being Cauchy in Gromov-Hausdorff metric, since if the GH distance between two spaces is $<\epsilon$ and one has a large $1$-separated subset, the other one must have a $(1-2\epsilon)$-separated subset of the same cardinality.
For a concrete example, take a sequence of open disks $D_n$ on the sphere $S^2$ with disjoint closures, and begin attaching a "needle" of length $3$ to each disk. Let $M_n$ be the space with $n$ such needles attached. The diameter of $M_n$ remains bounded by $10$. Also, if $m\gg n$, the distance $d_{GH}(M_m, M_n)$ cannot be small: there are only so many points at distance about $6$ from one another that one can fit into $M_n$ for a fixed $n$.