How long can human tissue last without blood?
As EMT (Emergency Medical Technician) instructors will tell you now, when wearing a tourniquet, you have about four hours to get the injury to a vascular surgeon and get it cared for appropriately. The rule therefore is four hours. However, the common thread I see in most sources is that about an hour and a half is the longest you want to go without blood flow to a limb.

As far as what happened to you, it sounds more like the limb simply fell asleep. When a limb falls asleep, it is getting the nerve pinched off for a time as well as loosing some blood flow. However, this is not a complete loss and so your arm is able to survive because it is still receiving enough oxygen and other nutrients to sustain its existence (as well as having toxins and CO2 pumped away from it).  

I had a very difficult time finding any concrete, scientific information on this subject overall.

So, if you would like more information about this, please visit the following links. The first is a WikiPedia link on Limb Infarctions (or an embolism in a limb). The second is from Mental Floss on Why Limbs Fall Asleep. The third is a somewhat random article on someone's personal experience on Sleeping on their arm.

So, in summary: although I could find no solid evidence on how long a limb can live without its blood supply, it appears to be about 2-4 hours (some sources say 5-8); however, I would like to add that your case appears to be a severe case of an asleep limb, you were/are likely in no danger. Also, if your arm indeed did have circulation completely cut off, it would have turned pale, then blue and later a livid red and purple as the blood began to pool; you then would begin to experience the effects of the limb dying i.e. necrosis, gangrene.