Yep, this has happened to me countless times, though usually it's only like my left arm (I lay on my left side most of the time). The other day, my left arm was so numb, I started pinching it with my right hand - needless to say, I didn't feel it, but it was just weird
I would say try sleeping in a different position, but when someone sleeps, they're not in control of their twists and turns in bed.
Though the only advice that I can make is try to take some Aspirin before lying down. Aspirin thins the blood, so it can more easily flow through your vessels when you have some kind of pressure on them. Though if you know that you've had adverse effects when taking Aspirin or some other blood thinner, skip it.
If this continues to bug you, just ask your doctor if there's any solution next time you go to visit.