Technically you could. However, this is actually incredibly rare, since you need to be ovulating to get pregnant. While menstruating, your body is expelling the materials of a failed pregnancy — your ovum, uterine lining, etc. That doesn't mean you still shouldn't take precautions if you aren't trying to have a baby. It's best to keep birth-control measures consistent.
However, if you've found a rare man that isn't freaked out by sex at that time of the month, lucky you! If you do decide to have sex during that time of the month, you can do what many couples do to save muss 'n' fuss: have sex in the shower. But others simply drape some towels over the bed linens so they feel free to get up to whatever they want.