We were given a flight safety booklet, with many examples of rejected T/O's above V1 resulting in overuns. It is very interesting to note that in EVERY case, had they continued they could have safely become airborne.
re: Quids Post, we give a very specific brief saying, "once thrust is set we will ONLY abort for an engine fire, failure, or something affecting the safety of the aircraft (to fly)".
If in your own mind you are not sure whether to stick with this, then you are very much leaving yourself open to a abort leading to an accident. Most aborts above V1 were due to sensory inputs eg) loud bangs (tyres etc).
You must be in a mind set to resist the temptation to stop for this alone. If tyre(s) have burst then you MAY NOT stop in time even if you abandon BELOW V1, as you do not have full braking capacity (assumed for the calculations).
You are almost always better continuing, sorting the problem out, returning for a full length landing with fire trucks etc. eg) follow your brief.
I don't know about you, but I tend to call V1 about 5 kts BELOW V1, as any problem which occurs above V1-5 CANNOT be assessed and a stop be ACTIONED by V1 (I fly an aircraft with rapid acceleration) . This is especially true if V1 coincides with Vr - if you call "VeeWun,Rotate" at V1/Vr then by the time you've said that and the other guys reacted you're about 10kts past Vr - this starts to impact on your climb and obstacle clearance performance. Don't forget that it is ME who is calling any potential failures, so obviously I will only call V1 at V1-5 IF a major failure has not already occured. I believe this is a practice that is approved in several flight safety quarters.