We prove that if a family of metrics, gi, on a compact Riemannian manifold, Mn, have a uniform lower Ricci curvature bound and converge to g∞ smoothly away from a singular set, S , with Hausdorff measure, Hn−1(S)=0, and if there exists connected precompact exhaustion, Wj, of Mn∖S satisfying diamgi(Mn)≤D0, Volgi(∂Wj)≤A0 and Volgi(Mn∖Wj)≤Vj where limj→∞Vj=0 then the Gromov–Hausdorff limit exists and agrees with the metric completion of (Mn∖S,g∞). This is a strong improvement over prior work of the author with Sormani that had the additional assumption that the singular set had to be a smooth submanifold of codimension two. We have a second main theorem in which the Hausdorff measure condition on S is replaced by diameter estimates on the connected components of the boundary of the exhaustion, ∂Wj. This second theorem allows for singular sets which are open subregions of the manifold. In addition, we show that the uniform lower Ricci curvature bounds in these theorems can be replaced by the existence of a uniform linear contractibility function. If this condition is removed altogether, then we prove that 18c9e8782ce385cc5c4778d2c1">, in which and N′ are the settled completions of c2093258db38696f1f2e7b6c26bd5" title="Click to view the MathML source">(M,gj) and (M∞∖S,g∞) respectively and dF is the Sormani–Wenger Intrinsic Flat distance. We present examples demonstrating the necessity of many of the hypotheses in our theorems.