No, it's not normal, really. The only time they would hurt is when going through puberty in your teens but other than that they shouldn't hurt.
Cindy you did not give your age, but I had bouts of breast tenderness during the phases of primenopause, and menopause. I have had everything checked out, which I suggest you do, but that is a possible explanation. Hormone level changes can cause these type symptoms.
I'm 35 and my breasts hurt to touch even lightly. It hurts to put a bra on or to lie on my stomach at night. They hurt all the time and sometimes they are red. Should I be concerned?