If you cut off the circulation to a ligament, can it become really bad?

Sometimes while i'm sleeping, because of a position i'm in, my whole arm will "fall asleep" and feel rather detached - you know the feeling. I understand why it does, but I'm wondering: what if it took me a really long time to wake up and catch it? Could I be causing myself real bodily harm? Ex: this morning when I tried moving it to get the blood flowing again, i had to swing it with my other hand and i could feel the life coming back as it went down my arm and then my fingers felt really cold. Are there lasting effects of this?