LMS, I don't think that's correct.
BAbies get antibodies from their mothers - that's immunity to whatever mother has antibodies for. It's not 100% for many things though, especially colds where the virus mutates so readily. However, there is no other way for babies to get immunity in this way when so young, as their immune systems are so immature. As they get older and their immune system matures, they will develop their own immunity to whatever infections they fight off.
There are lots of factors involved when we think about frequency & duration of illness - exposure is a big one. With an older sibling, your baby is much more likely to be exposed to seasonal bugs (and the time of year will be a factor here too). Your antibodies will be doing a sterling job of helping your little one fight off infection, however, and she would not be getting this at all if fully formula fed.
My DS didn't get his first cold till 11 mnths. My daughter has had several and other viruses as well through this winter - it was a bad one this year, I think, and DS was in daycare so unfortunately she copped it. She didn't ever get as sick as DS, however, and I'm sure that's down to the antibodies she was receiving.