Let $f:X\to Y$ be a pointed map of pointed connected $n$-dimensional CW complexes. Whitehead's theorem says that if $f_*:\pi_qX\to \pi_qY$ is an isomorphism for $q\le n$ and a surjection for $q=n+1$, then $f$ is a homotopy equivalence (e.g. Theorem (Whitehead) on p.75 of May's "Concise Course in Algebraic Topology").
I am interested in counterexamples to this when you drop the surjectivity condition for $q=n+1$. That is,
Question: What examples are there of a map $f:X\to Y$ of pointed connected $n$-dimensional CW complexes that induces isomorphism on $\pi_q$ for $q\le n$, but is not a homotopy equivalence?
I would also like to know what are the "minimal" examples of this. For example, it seems impossible for $n\le 2$ (the induced map on universal covers is a homology isomorphism by the Hurewicz theorem, and hence is a weak equivalence and thus induces isomorphism on all higher homotopy groups). I also wonder if there is an example with finite complexes.