When can the $\mathcal Z$-transform be inverted? When not?
If I understand your question correctly then you're asking under which conditions the inverse $\mathcal{Z}$-transform of a given function $F(z)$ exists. Since the inverse $\mathcal{Z}$-transform is just a Laurent expansion of $F(z)$ around $z=0$, the inverse transform exists if $F(z)$ can be developed in a Laurent series about $z=0$. This is the case if $F(z)$ is analytic in an annulus centered at $z=0$.