When Stephen Smale was a graduate student, he thought he had a proof of the Poincaré Conjecture as follows: Take a compact simply-connected 3-manifold M and remove the interiors of two disjoint 3-balls to get, say, M1 having as boundary two copies of S2. It is easy to show that M1 has a nonsingular vector field entering along one S2 and exiting along the other. Clearly by the simply-connectedness of M, each orbit entering on one boundary component must exit on the other one. Thus M1 must be S2 x [0,1] and hence by replacing the removed 3-balls, M must have been S3. QED.
I'm not sure who first pointed out the error, but undoubtedly understanding examples like this helped him appreciate the subtlety of the problem and ultimately prove the Generalized Poincaré Conjecture for dimensions ≥ 5.