I'm wondering why can't one take a blood pressure or draw blood on the arm of the side that a mastectomy was done?

Breast ca sx. After breast ca, sx there is changes in blood flow drainage especially if there was axillary nodes removed that will inter fer with lymphatic drainage . You will be at high risk for phlebitis and lymphangitis.
Arm swelling. There is lymphatic interruption with radical mastectomy -- so frequent and prolonged constriction with BP cuff may cause swelling and pain in the affected arm and hand.