Is it possible to perform a connected sum of two Riemannian Manifolds or Orbifolds while keeping curvature bounded from below? More explicitly, If $M_1$ and $M_2$ are two Riemannian manifolds (or orbifolds) of (sectional or Ricci) curvature bounded from below, is it possible to obtain a Riemannian metric in $M_1\#M_2$ that also has curvature bounded below?
It is possible to take the connected sum of manifolds of dimension at least 3 of positive scalar curvature, and get a metric of positive scalar curvature, by the work of Gromov and Lawson,
Gromov, Mikhael; Lawson, H. Blaine, Jr. Positive scalar curvature and the Dirac operator on complete Riemannian manifolds. Inst. Hautes Études Sci. Publ. Math. No. 58 (1983), 83–196 (1984).
For sectional and Ricci curvature this is not possible. Of course, you get some lower bound even in these cases by compactness. If you have other conditions in mind you need to make your question more specific (and in particular scale-invariant).