An elliptic space is one whose rational homotopy and rational cohomology are both finite dimensional. David Anick conjectured that any simply connected finite CW-complex $S$ can be realized as the $k$-skeleton of some elliptic complex as long as $k\,{>}\,\dim S$, or, equivalently, that any simply connected finite Postnikov piece $S$ can be realized as the base of a fibration $F\,{\to}\,E\,{\to}\,S$ where $E$ is elliptic and $F$ is $k$-connected, as long as the $k$ is larger than the dimension of any homotopy class of $S$. This conjecture is only known in a few cases, and here we show that in particular if the Postnikov invariants of $S$ are decomposable, then the Anick conjecture holds for $S$. We also relate this conjecture with other finiteness properties of rational spaces.