How about after a certain amount of time the tournament starts irregardless of how many people have accepted the invite? You fill the holes with byes or AIs.
This idea is great, except the AI part, which is pointless. I'm pretty sure, it will easily fix the current situation. Unfortunately it only works perfectly for 1v1 games. In case of team games, it would be annoying to fill AIs in the missing slots. Instead you could fill uncomplete teams with players from other uncompleted teams and kick those players that are left over, starting with those, that have joined the tournament at least. Or just kick those uncompleted teams.
In my opinion, players, who join team game tournaments are responsible to fill the missing slots in their teams. If they simply blockade an entire team for theirselfes, they are part of the problem as well.
Edit: tournaments would need a countdown, that shows when they will be enforced to get started
Edited 2/10/2015 12:23:08