It hasn't happened to me, but it did happen to my opponent in one
game where he just handed over his queen. He said that he had
moved the piece there originally, saw the error, then moved another
piece instead. However, when he hit the move button, his original bad
queen move was taken. In this case, we declared the game a draw
and started over, since it would have been a pointless game once the
queen was erroneously blundered away like that.
If I've been moving various pieces around "experimentally," then I
always hit the "refresh" button in the browser before making my final
move and pressing the move button. So far, I've never had a problem
with the system picking up the wrong move.