Whilst unlikely, could they not sue for damages? For instance, perhaps they had made commercial commitments on the basis that you would be joining, which they now find themselves unable to deliver and thus having to back-fill using expensive agency staff?
Is an employment contract different to any other contract? Usually, a breach of contract means that, if one side is able to demonstrate that the breach has been to their detriment, they can recover the costs from the other party? In which case the way out of having already signed an employment contract would be to hand in your notice on day 1 and then work through your notice period unless mutually agreed otherwise?
If it's a big company, I expect however that they would just be used to this sort of thing happening and would be cheesed off and swear never to deal with you again - but that would be the end of it.
(This isn't legal advice by the way, just some random thoughts!)