Pressurisation 101.
The pressurisation and air conditioning systems are basically the same system.
More air is pumped in than leaks out. Aircraft are not the perfect pressure vessels that some think they are. Turn the pressurisation off at altitude, and close all the valves, and the cabin pressure will drop fairly quickly. At a guess...about five hundred to two thousand feet per minute.
Outflow valves control the amount of pressurisation. Basically they are little more than a hatch that opens to the outside. If they open they let more air out, closed less. Aircraft have various numbers of these valves. An A380 has 4, a 747 has two, and 767 has one. I'd expect the 737 would also use one.
A pack is the air con/pressurisation unit itself. Again the numbers vary. A380 has four, 747 three, and 767 and 737 have two. Failure of an individual unit has little effect, and is allowed within the MELs, although with the twin installations that may include an altitude restriction (something around FL350 or so).
Ok, so what can go wrong with pressurisation? Well, the most spectacular, and least likely, is some sort of hull rupture. In that case you are looking at cabin rates of climb in the order of 60,000 feet per minute. This is extremely rare.
Most commonly a pack will fail. In itself this will have no effect, as the other(s) will just work a bit harder. The crew may need to descend slightly (only in aircraft with only two packs), though this should not be noticeable in the cabin. There is no urgency to this. A fairly common event.
Controller failure. Normally the system will just switch/be switched to an alternate controller, with no further action required. The ultimate failure mode here is manual control, which is quite adequate. PITA for the crew though.
So, the most likely reason that you'll need to descend to 10,000 feet. Well I'd suggest the loss of both packs. Now, this won't be a one pack fails, followed by the other scenario, but rather a dispatch under the MEL, and then a failure of the good pack in flight. You then have no pressurisation, and need to get to 10,000 feet. Is there any danger? No, none at all (unless you do a Helios, and stay up there). The aircraft will slowly lose cabin altitude during the descent (because it isn't a leak proof vessel), and you may get to 10,000 feet before the masks auto deploy. But, if you are at an altitude that gives you a high cabin already, you may only have (say) 5,000 feet of cabin altitude before auto deployment. Couple that with an aircraft that leaks rapidly, or a descent that is delayed by procedures/traffic, then you may end up with the masks deploying, even though they really aren't needed.
