My first baby does not get sick for the first year but he is on formula, however he get sick so often since he started school.
My second child is breastfed and she first get sick when she is only 2 months old because her brother get sick. She get sick each and every time her brother get sick but when she started school, she get sick less often.
My third child is now 8 months and she drinks breast milk also. She got sick when she was 6 months and that's when I got sick.
So conclusion is that so long there's germs in the world, baby will get sick, regardless they drink formula or breastmilk. However, the antibody will help prevent the child from getting sick again from the same germs. And these antibodies come from breastmilk AND previous exposure to the germs, ie., getting sick.