It is my understanding that after a certain point the takeoff cannot be aborted. If the tire did blow, and the crew didn't realize it until the last minute, it seems that they would have gone ahead with the takeoff.
It's a complex decision that almost has to be prepackaged, i.e. you have to train for likely events and make the right call, there isn't time for much analysis while the runway is disappearing behind you. A rejected takeoff above 80 knots in an airliner is normally only done in the case of engine fire, engine failure, loss of directional control or if the aircraft is unable to fly. In some airlines, either pilot can call for a rejected takeoff, for most airlines it is the captain's call in my experience. Of course, this mishap was a bizjet, not an airliner and as mentioned in an earlier post, some outfits have 'co-captains' where everybody has four stripes (been there, done that myself, didn't really like it).
A good article on this subject with recent (a decade ago) regulatory thinking, some statistics, and accident data starts on page 3 here:
http://www.flightsafety.org/fsd/fsd_oct98.pdf
______________________________
We used to arm reversers on takeoff until reaching V1 speed.
I sure don't remember that one, I used to fly Lears with Dee Howard, Aeronca, and no T/R's. Was it a toggle switch or something? It's been a long time. --> Just reread your post and answered my own question...