Along the same lines: you can produce examples involving simply connected finite CW-complexes, by using S1$S^1$ actions: CPm x S2n+1$\mathbb{C}P^m \times S^{2n+1}$ and S2m+1 x CPn$S^{2m+1}\times \mathbb{C}P^n$ have the same homotopy groups, for instance.
It's easy to produce examples with arbitrarily high connectivity: take the homotopy fiber X$X$ of any non-trivial map K(A,m)->K(B,n+1)$K(A,m)\to K(B,n+1)$, with n>m$n>m$. Then X$X$ and Y=K(A,m) x K(B,n)$Y=K(A,m) \times K(B,n)$ have the same homotopy groups, but are not homotopy equivalent.
Here's a new question: for given k$k$, can you find a pair of k$k$-connected finite CW-complexes which have the same homotopy groups, but aren't homotopy equivalent?
