I think it depends entirely on your university. We were told that unless we're dead there are no mitigating circumstances. Even if you're ill or injured or whatever before the deadline, you should have been working on it all year and so it shouldn't really effect it that much. Rather specifically, we've been told if we've broken a leg and come deadline day we're stuck in hospital, get a friend to submit it for us. Of course, that all sounds a little extreme, but basically for my department at my university, it would require a serious case to be given mitigating circumstances with the argument of "you shouldn't have left it so late".
Perhaps talk to your supervisor, let them know of your concerns and the cause of them to see if they think it would be worth submitting a note. They'd be much more experienced with the administration side of things for your department and university than us.
EDIT: I just want to make it clear, I'm not being judgmental by saying you should have done it by now and this shouldn't have much of an impact. This is the attitude my department has taken towards 'excuses'. I've done like half of mine so far with no excuse for not having done more, myself.