Two pointed, connected CW complexes with the same homotopy groups need not be homotopy equivalent (Are there two non-homotopy equivalent spaces with equal homotopy groups?). Moreover, having the same homotopy and homology groups is also not enough (Spaces with same homotopy and homology groups that are not homotopy equivalent?).
Question: Suppose that two pointed, connected CW complexes have homotopy equivalent $n$-Postnikov sections for every $n \geq 1$. Are the spaces homotopy equivalent?
I expect the answer to the question to be negative, but I'm having a hard time finding a counterexample. So far, I know that such a counterexample would have both spaces have the same homology and homotopy groups, and that both spaces would need to have nontrivial homology and homotopy groups in arbitrarily high dimension.