From an interview with Philip Wolfe:

[John von Nuemann] always contended with Dantzig that the simplex method would take an absurdly long amount of time to solve linear programming problems. It appears to be, oh, so far as I know, the one place where Johnny went very badly wrong.

Quoted in Undergraduate Convexity

Don’t judge. It took a new type of analysis (smoothed analysis) and a Gödel Prize to prove it works!

I’d be happy if someone could say I was only really wrong about one thing!

To be fair, the naive simplex method that he was familiar with now does look interminably slow by today’s standard. A competitive modern simplex implementation requires an enormous amount of algorithm engineering technology that was developed since the 1980s (not to even mention increases in hardware performance improvements).

He also suggested to physicists that hidden variable theories of quantum mechanics were not possible despite the fact that they are, e.g., Bohm’s theory. A recent analysis of J. Bell vs. J. von Neumann on this matter by J. Bub can be found at http://arxiv.org/abs/1006.0499

If von Neumann was talking about worst-case performance, then he was spot-on.