I’ve seen it in Robin Hood; I’ve seen it in The Good, the Bad, and the Ugly: Someone is about to be executed by hanging, and someone saves them by shooting the rope with a bullet or an arrow. Has this ever successfully been performed in reality? Is it even theoretically possible? Has Mythbusters ever attempted this?
Honestly I don’t have any proof of this but anything short of a shotgun with birdshot at close range would be useless. Any solid bullet would probably deflect or not completely cut the rope due to flexibility in the rope and multiple strands. A hunting arrow with a chisel head and multiple cutting heads might be better but even that from 10 yards or closer is a “miracle shot”.
I don’t know specifically about the case where someone is being hung by the end of the rope in a gallows scenario, but I’ve seen other examples of people shooting rope. Here is one (which is not particularly comforting if you’re relying on this method for someone to save you dramatically at the last second).
ETA: here is another (more comforting) example, with a much larger round and a much more skilled marksman
Gruesome thought : not much use if they were doing the “long drop” method
There are two main different ways of hanging someone. You can have little to no drop, in which case the person strangles to death. It’s a very long, unpleasant death, but is commonly used in the Middle East to this day. With this type of hanging, the person is alive and on the end of a taut rope.
Mythbusters and others have attempted to recreate this, and generally you’ll run out of bullets in your six-shooter long before the rope breaks. An old west rifle or a bow won’t be able to do it in one shot either. Given enough shots, you can theoretically cut the rope, but it’s not going to be quick.
The second type of hanging uses a longer drop, and if done properly, the victim’s neck breaks, and they have a relatively quick and painless death. By the time there is tension on the rope, it’s too late to save the victim, one way or the other. Most official hangings in the old west used this method.
Technically there is also a third case, which usually only happens when someone screws up the drop calculations and the body drops too far. In this case, if there is too long of a drop, instead of just breaking the neck, the head pops off. Most hangings try to avoid this due to it being rather gruesome for the spectators.
In any case, if you try to shoot the rope before there is tension on it, the cut from the bullet or arrow is going to be even less effectively since a lot of energy will be expended just moving the rope.
Not Mythbusters, but the show Hollywood Weapons attempted the shot it in S1E3, “The Good, The Bad and Terry” (available on Amazon Prime).
Terry couldn’t do it, but said if everything was perfect it might be possible
Real life teaches us that such precision shooting is commonplace.
It’s possible. A set-up like this was a regular feature on Top Shot. Shooters managed to cut the rope every time. If you gave them enough shots. 
They make (or used to make) 12 gauge shotgun rounds that were basically two metal balls with a wire in between them, the idea being in flight the balls would expand and the wire could then be used to cut whatever was in its way increasing the area covered by the shot. Wonder if that would be effective.
Maybe the myth originated in a misunderstanding from short/no-drop hangings? The trick isn’t to try and shoot out the rope, see, but to shoot whoever’s on the end of the rope and put them out of their misery Last of the Mohicans style.
Taking out the crossbeam with a .50 cal would probably work (although the person being hanged might then be injured by falling pieces of timber)
I’ve seen these sorts of rounds reviewed on the YouTube channel Taofledermaus. When you examine their flight under slow motion video capture, they typically don’t do what they are supposed to do - they either break apart, or clump up
Bolo shells. A modern take on amuch older idea. And banned in a lot of places.
Were they aiming at a non-moving rope? I’d think that the person being hung by strangulation would be moving, both from the rope swinging from the drop and from any struggling they might do before they lost consciousness. I suppose the rope would be moving less nearest to the crossbeam or whatever the noose is hanging from, but wouldn’t be totally still.