Leighton's Theorem: extensions, limitations, and quasitrees

Leighton's Theorem states that if there is a tree $T$ that covers two finite graphs $G_1$ and $G_2$, then there is a finite graph $\hat G$ that is covered by $T$ and covers both $G_1$ and $G_2$. We prove that this result does not extend to regular covers by graphs other than trees. Nor does it extend to non-regular covers by a quasitree, even if the automorphism group of the quasitree contains a uniform lattice. But it does extend to regular coverings by quasitrees.