For some problem in Algebraic Topology (presumably related to homotopy groups or similar, with free groups): I thought that two groups $G$, $H$ were isomorphic, because $G \approx A \subseteq H \approx B \subseteq G$, where $\approx$ means "is isomorphic to" and $\subseteq$ means "is a subgroup of".

However, I was very shocked when informed that this does NOT imply that $G \approx H$ in general! (I thought this was true, and spent a long time trying to prove it; but I knew that I hadn't succeeded. So I suppose this doesn't qualify, but anyway).