Can you get a blood infection from a blood test?

Very very rare: If the person drawing blood does not take usual precautions for preparing skin and wahsing hands, micro-organisms present on the skin may get introduced into the blood stream. I have not heard this happen in 40 years.

Very unlikely: The phlebotomist (person drawing the blood), will clean the area thoroughly with alcohol or iodine -- so the chance of an infection is really really low.