What could cause really bad breast tenderness so you have to wear a bra?

Fibrocystic changes. You could have fibrocystic changes. This is the most common cause of breast pain. It can be aggravated by caffeine, stress, menstrual cycle and hormonal supplements. If you are over 40, then you should have a mammogram and ultrasound. Please get evaluated by a breast specialist.
Many things. Periods, pregnancy, coffee, breast feeding, infection, lesions in breasts (cysts, lymph nodes) medications, etc...