Some people have asked if this would work with a person on the end of a long rope, and the answer is yes! Here's one way to think about it:

When the rope (or the slinky) is hanging there, every bit of it is sitting still, with the pull from the bit of rope above it exactly matching the pull from the bit below it, plus a tiny bit more for its own weight. Now when you cut the rope, the bit at the very top suddenly has no more pull from above. BUT, all the bits below are still pulling on each other just as before. So only the top bit starts moving at first. As it moves closer to the next bit, the tension between those bits decreases, and the next bit decides that it should fall too, and so on. Each bit of the rope has to wait for the bit above it to move closer, before it loses its pull from above.

So the question is, how long does it take for all that movement to happen? That might seem hard to answer, but it's actually more familiar than it sounds. For example, if you jiggle the top of the rope instead of dropping it, a very similar thing will happen. The top bit jiggles, and it causes the next bit below it to jiggle, all the way down. We've all seen this happen when we play with slinkies or swings. This is also very similar to how sound works: I clap my hand or something, and air molecules that jiggle make the molecules next to them jiggle, all the way out. So what we're basically asking is, what is the "speed of sound" in the rope? It turns out that that depends on the weight of the rope, and how much tension is in it. (http://en.wikipedia.org/wiki/Wave_equation)

So if you put a person on the end of the rope, that person probably weighs a lot more than the rope itself, so the tension in the rope will be a lot higher. That means the speed of sound will be higher too, and it will take less time for the cut at the top of the rope to jiggle its way down.

Another way to think about it is that gravity is still working, so if the rope is holding up the whole weight of the person, then the top of the rope will be pulled down with all that weight. But the rope is pretty light, so it will snap down very quickly, and it won't take very long at all for the person to start falling. But just like the slinky, it does take some time!

In a sense, this is also how planes and helicopters work. Instead of falling down themselves, they push a lot of air down, and that lets them fight gravity. The bottom of the slinky is doing the same thing, except instead of pushing it's pulling, and instead of air it's just got more slinky :)