When the Impossible becomes Possible again

Many times mathematicians draw proofs about the impossibility of something. As an example, take the Abel–Ruffini theorem, which states no generic formula exists for quintic equations (I know the proof is more contrived that this).

Some "proofs" might be wrong though: maybe a formalism was overlooked... maybe there was a failed axiom. What I want to know is examples of once claimed proofs on the "impossibility" of something, but which later were revealed wrong?

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If a proof contains a mistake, it won't take mathematicians long to spot it. If a proof is sound, it will remain eternally true within its set of axioms. This is one of the beauties of Mathematics. – Ayman Hourieh May 6 '12 at 21:48
Hence... Are there examples of something that was kept too long before someone spotted it? :-) – Hugo S Ferreira May 6 '12 at 21:50
@AymanHourieh Not always true - see my answer. – Bill Dubuque May 6 '12 at 22:00

Perhaps one of the most famous is Cantor's $1887$ "proof" of the impossibility of infinitesimals. This was presented in a letter to Weierstrass. The "proof" was later elaborated by Peano $(1892)$ and Russell $(1903)$ in their arguments against infinitesimals. Of course, nowadays, we know perfectly rigorous presentations of infinitesimals thanks to the groundbreaking work of Abraham Robinson (nonstandard analysis), as well as alternative approaches such as Synthetic Differential Geometry (which employs nilpotent infinitesimals).