First realize that the right definition of orbifold is build on topos (the definition without topoi are mostly cheating).

Here is an example where similar things appear as limit of Riemannian manifold. [For more info, check "On the long-time behavior of type-III Ricci flow solutions." by Lott or the appendix in paper "Diffeomorphism Finiteness, Positive Pinching, and Second Homotopy" with Tuschmann.]

Consider the circles $\mathbb S^1_\ell$ of length $\ell>0$. As $\ell\to 0$, in Gromov--Hausdorff topology they converge to point. On the other hand, the space of harmonic 1-forms stay the same on $\mathbb S^1_\ell$, so for some problems Gromov--Hausdorff topology is not right choice.

You may think of $\mathbb S^1_\ell$ as $\mathbb R/(\ell{\cdot}\mathbb Z)$. Then it is natural to think that $\mathbb R/\mathbb R$ is the limit of $\mathbb S^1_\ell$; More precisely $\mathbb R/\mathbb R$ has one chart $\mathbb R$ and all the shifts $x\mapsto x+a$ as the transition maps. So the limit is smooth, but not a manifold. You may think of it as a pseudogroup of transformations, but you need topoi to give an invariant definition (the same story as with orbifolds).