In 1921, Emile Borel claimed that the mimimax theorem is false for symmetric zero-sum games with a finite number of actions greater than 7. He didn’t see the need to actually give a counterexample, since “it is easy to see that, once n exceeds 7,” there may be no mixture holding the other player to 0. Oops. This makes me feel a little bad about asking people to prove the theorem on the qualifying exam, but hey, we did go over the proof in class.
For the curious, a translation by the well-known translator L. J. Savage was printed in Econometrica, Jan. 1953, 97-100. While giggling at the error, we also see that Borel gets credit for describing iterated dominance and mixed-strategy equilibria well before Von Neumann-Morgenstern.