If it's the blood of someone who's HIV positive - then yes.
Actually, you can if the blood is HIV positive, hasn't been exposed to the air for an extended period of time, and you have an open sore anywhere in your mouth or esophagus. If not, then the risk is minimal. HIV cannot survive in the air for very long at all. Also, the digestive acids found in your stomach will destroy the HIV. There's actually a greater risk of contracting Hepatitis. Hepatitis can live outside the body for up to 2 weeks and can be transmitted via the saliva. Personally, I don't make a habit of swallowing blood and advise against taking the risk.
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Our digestive system is too acidic for HIV to survive, so NO, you won't get HIV from swallowing HIV-infected blood. BUT if you have cuts/ulcers in your mouth/throat/stomach, then you can be infected. Just eating food will create tiny cuts in your mouth, so be VERY careful if you want to drink blood-for whatever reason :)