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The general idea is that whatever mechanism is allowing the ropes to move is something that 'fails closed'. The simplest mechanism is to wrap two prusiks[1] around the rope that is moving. Those prusiks are then securely attached to an anchor or some sort. Left alone, the prusiks will 'grab' the rope and prevent it from moving. Someone has to be actively holding them in position to allow the rope to move (otherwise they'll slide with the rope and pull taught).

In other cases, it's a question of how you set up things like descenders[2]. The same basic principle applies though. You need to be actively holding it open in order for the rope to move through it. There are a few exceptions to this (things like figure-8 descenders[3]), but in those cases, they are being used by a single operator to control their own descent.

The most common situations where this goes wrong are during transitions. The system used to lower someone over the edge is different from the system used to haul them back up. This exercise is fairly simple, but it requires a great deal of attention to detail, and the steps have to be done in the proper order to ensure there are no 'gaps'.

The number of people who would fall would depend completely on the operation. Generally speaking you have a single medic over the edge, who may have the victim clipped in to them if they're on the way back up. In some cases, you may have one or more additional rescuers.

[1] https://en.wikipedia.org/wiki/Prusik [2] http://www.petzl.com/en/Professional/Descenders/RIG [3] http://www.cmcrescue.com/equipment/rescue-8/