For $d=3$ the homotopy groups can be pretty elaborate. Consider the connect-sum of some lens spaces. The universal cover embeds in $S^3$ as the complement of a cantor set (except for a few degenerate cases where you have $\mathbb RP^3$ summands). So the homotopy-groups are pretty complicated ($\pi_2$ is finitely generated over $\pi_1$). You could probably make an argument that this is about the worst thing that can happen for the homotopy-groups of 3-manifolds.
You might want to phrase your question as a question about the Postnikov towers of manifolds. Eilenberg-Maclane spaces are rarely compact boundaryless manifolds.