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.
Is It Normal For A Womans Breasts To Hurt All The Time?
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?