Is it possible that two topological spaces share all higher homotopy groups, but are not homeomorphic? I should note that I have not studied much in the way of the theory of higher homotopy groups; I merely know of the groups' existence, and was wondering what could be said, in broad terms, about such a case.
If such a case is possible, an example would be appreciated.
Yes it can happen. A simple example is to take $X=S^1\vee S^2$ and $Y$ to be a circle with a two spheres attached each at a different point. Then $Y$ is a double cover of $X$, so it shares all higher homotopy groups $\pi_n$ $n\geq 2$. (Here we use the fact that a covering map induces an isomorphism on all higher homotopy groups.) Also $\pi_1(X)=\pi_1(Y)=\mathbb Z$, so $X$ and $Y$ have the same fundamental group as well.