The first is called "Dehn's lemma", concerning when knots in 3-space
are trivial. Dehn published a "proof" in the 30's that was accepted for
10 years or so I believe, then it was seen to be incomplete and the
proof was finally completed by Papakyriakopoulos (spelling probably
wrong) in the 50's. There's a recent article in the "Mathematical
Intelligencer" on Dehn that probably has details.
A second is the so-called "Hard Lefschetz theorem" on the cohomology of
algebraic surfaces. Lefschetz actually got a prize for a monograph in
which this was the crowning result (in the 20's). It became rather slowly
clear that no one could understand his proof, but we had a seminar in
the 50's trying to read it, before we gave up. It was proven by Hodge
by totally different methods in the 40's.
Both individual mathematicians and communities of mathematicians in various
subfields can and do fool themselves about what is a proof from time to time.
David Mumford