Can a New mattress have bed bugs?

I just got a new bed (mattress, box spring and base). I woke up today with 6 bites on me that itch like mad and that (without scratching) have grown to large welts. Is this a sign of bed bugs? The mattress is new, never used. Is it possible that I brought bed bugs in with a new mattress? Don't they disinfect mattresses/beds before they are sold?