I've never had that problem but I have had it glitch so that one of my pieces is placed somewhere randomly on the board and I can't move it at all...even if I try to move it from it's starting position. This is particularly annoying because I can't move any pieces through where that piece should be so this means I normally can't castle that game.
The best chance to fix it is by refreshing though that doesnt always work.
I was playing live chess, and I made my move. Then my opponent moved, and this happened. I couldn't move any pieces. I thought the chess.com staff might want to know about this.
EDIT: As you can see, my clock still counted down; it counted as if it were my turn.