Technically, in the context of the historical Old West, you probably couldn’t, but under controlled conditions, you obviously can, as @stranger_in_a_strange_land and some guy on the History Channel pointed out.
Let me explain why.
A rope used for hangings in the Old West would probably be about a half inch or so thick and made from braided twine. If the man who was being hanged did not suffer a broken neck as soon as the rope extended after the trapdoor dropped, he would only have a few seconds before he suffocated. During this window of time, his partner, armed with most likely a Winchester lever-action rifle, chambered in .45 Long Colt, would have to fire off enough rounds (about 1 shot per 2 seconds) to split the rope from a relatively close range, because this shot would be incredibly difficult from more than a few yards away, even with a quality rifle with open iron sights. During this time, the guy with the rifle would probably be shot or zerg rushed by the crowd for conspiring to rescue a condemned man. That’s bad juju, ya know?