The Buffon's Needle method converges to pi very slowly. (There's a statistical method for estimating how slowly, but I don't know it; maybe somebody else does?) There was an incident where someone claimed to have done the experiment (I think while in a hospital for a while) and gotten a decent value, and mathematicians later rejected the claim as implausibly accurate!

One gimmick that people have mentioned is that if you don't choose ahead of time how many trials to do, you can stop the experiment when it looks like the value so far is near what you know pi should be. That allows some people to say "I did 47200 trials and got a pretty good value!", but it doesn't mean that they would have gotten such a good value without choosing when to stop.