Well, anything is possible. But it would have to be really bad circulation. And a really deep sleep. If your circulation is otherwise normal (ie: no neuropathy), then it takes more than just sleeping on your arm to cut off the blood supply long enough to kill the arm. Most people don't sleep perfectly still: they move around at least a little bit. So if your arm is in a position where the circulation is bad, natural movement will ususally take care of it. In order to cut off the circulation enough to 'lose' an arm, you'd need the equivalent of a tournicate.
Anything that is going to kill your arm is normally enough to also wake you up. While it is true that restricting the blood flow just under the skin can cause a limb to 'fall asleep', it takes more than that to kill a limb. Losing blood flow to an arm hurts: the pain would wake you up. Also, if something is pressing on your arm hard enough to cut off the blood flow, then it's also probably pressing hard enough to wake you up. People just don't sleep through that kind of thing.
There are anecdotal reports of people who pass out while drunk in a weird position, and lose thier arm or leg. But being in a drunken stupor is not the same as being asleep. And they are usually in a position that they would not normally be in for any period of time. Now diabetics sometimes lose thier feet, but thats because the nerves in thier feet stop working - only somewhat related to circulation.
