Throughout, let $X$ be a *connected finite* CW-complex.

Question:If $X$ is of dimension $n$. Is there some integer $n'$ (maybe depending only on $n$), such that all homotopy groups $\pi_k(X)$ for $k \geq n'$ are finite?

For the spheres $S^n$, $n'=2n+1$ works by Freudenthal's Suspension Theorem and Serre's result that the stable homotopy groups in that range are finite. More generally, if $\pi_1(X)=0$, then the Milnor-Moore theorem relates the rational homotopy groups to the rational homology of the loop space of $X$ and I believe that this can be used to get a similar conclusion. But what if $\pi_1(X) \neq 0$?

**EDIT:** Igor Belegradek (besides answering the question) pointed out that what I stated in the last three lines is not correct.