DannyC; you are correct, many people would guess momentum. Remember, however, that momentum (mass x velocity) is directly porportionate to the velocity.
Energy, however (mass x velocity squared /2) is proportional to the square of velocity; this geometric progression is what allows the rapidly moving blade to cut through the rubber band while its' intertia holds it in place, before its' elastic tension can accelerate it out of the way.
Remember that any edged object works because it concentrates force in a very small area, over a very small TIME.
Note, if you please, that even if the tension of the rubber band is increased, the band still must start accelerating from a standstill, thus obviating most of the advantage of the increased tension.
Actually, I have no idea what in the heck a rubber band test is. However, if KuKu can cut a stretched rubber band into six pieces, he must work at Benehana! Walt