Suppose a metric space $X$ is the Gromov--Hausdorff limit of a sequence of Riemannian three-manifolds $M_i$ with Ricci curvature bounded from below and not collapsed. Is $X$ a manifold? What references are there concerning this problem?
This does not answer the question since lower bound on Ricci curvature is replaced by a stronger condition of lower bound on sectional curvature. Nevertheless perhaps it might be relevant.
Edit: By the Perelman stability theorem, if a sequence of smooth compact connected Riemannian manifolds $M_i$ has uniformly bounded from below sectional (rather than Ricci) curvature, converges to a compact metric space $X$ in the Gromov-Hausdorff sense, and for Hausdorff dimensions $\dim X=\dim M_i$ for large $i$, then $X$ is homeomorphic to smooth manifold (with non-smooth metric in general, of course).
For the Perelman stability theorem see p.2-3 here https://www.math.psu.edu/petrunin/papers/alexandrov/perelmanASWCBFB2+.pdf .
Added: May I add another statement which is a combination of the above Perelman theorem and Burago-Gromov-Perelman theorem (see Corollary 10.10.11 in the book "Metric geometry" by Burago-Burago-Ivanov).
Let a sequence of smooth compact connected Riemannian manifolds $M_i$ has uniformly bounded from below sectional curvature and converges to a compact metric space $X$ in the Gromov-Hausdorff sense. Assume in addition that for some $\delta>0$ one has $vol(M_i)>\delta$ for all $i$. Then $X$ is homeomorphic to $M_i$ for large $i$.