$(A,\mathfrak{m})$ a Noetherian local ring, $M\neq 0$ a finitely generated $A$module. As I understand, $\mbox{Ext }^{j}(A/\mathfrak{m}, M) = 0$ for $j<\mbox{depth }(M)$ and for $j>\mbox{inj. dim }(M)$, while $\mbox{Ext }^{j}(A/\mathfrak{m}, M) \neq 0$ for $j=\mbox{depth }(M)$ and $j=\mbox{inj. dim }(M)$. And I cannot help but wonder if $\mbox{Ext }^{j}(A/\mathfrak{m}, M) \neq 0$ for every $j$ between $\mbox{depth }(M)$ and $\mbox{inj. dim }(M)$ ?

Yes. See Fossum, Foxby, Griffith, and Reiten, "Minimal injective resolutions with applications to dualizing modules and Gorenstein modules" (Theorem 1.1) and also Roberts, "Two applications of dualizing complexes over local rings". An earlier paper by Foxby, "On the mu_i in a minimal injective resolution" settles several special cases, including when $A$ or $M$ is CM, $\mathrm{depth} M \geq \mathrm{depth} A$, or $M$ has finite injective dimension. 


This is not exactly the same but anyways it is related and very cute... In your context, given a sequence $n_0<n_1<\cdots<n_k$ of nonnegative integers, there exists a local ring and a module such that $\mathrm{depth}R=n_0$, $\dim R=n_k$ and the local cohomology groups $H^\bullet_{\mathfrak m}(R)$ is nonzero precisely in the degrees $n_i$; see [E. G. Evans Jr. and P. A. Griffith, Local cohomology modules for normal domains, J. London Math. Soc. (2) 19 (1979), 277–284.] 

