Barit1, if the runway is long enough and the aircraft is a STOL or nearly STOL type, or even a high performance type operating at a light weight 'V1 Max' for the situation described could be any speed that allowed it to stop by the far end. But then, in the interest of handling, tyre speeds etc Vr would happen before your 'V1 Max', so that opens up another can of worms and one which has never been addressed by certification.
My belief, of course, is that if it won't fly, I hope I recognise the fact in time to chuck it back on the ground. If it will fly, I will respect V1. It's the grey area between knowing and not knowing whether it will fly that is the problem. If seriously in doubt, on the ground is the better place to be, as long as you are not going to hit a bridge or brick wall or block of apartments on the over-run. Had the Concorde crew known what was unfolding, I bet they would have rejected even though they had passed V1. Sadly, they had no way of knowing just how bad it was.