I agree with most of what the previous posters say, with a few comments.
To call yourself a pediatric anesthesiologist, you need the fellowship.
What your medical center does for peds case types depends on several things. But at a major university hospital, it would be highly rare to escape from a regular anesthesiology residency without taking care of the sick kids. Most major universities have children's hospitals, hence is the dumping ground for the sickest kids-because the community physicians/hospitals tend to refuse these cases. These cases are for pediatric specialists of all types, including pediatric anesthesiologists, and unless you are doing the fellowship, chances are you won't be doing the Fontan Procedure, or a TAPVR repair (pedi heart surgery) in regular private practice.
In general, at OU, it seems the CRNAs get the tonsil surgery, and the residents here get stuck in the heart room, or the spine/neuro, or the major ENT cases.
Some might argue that unless you are doing the fellowship, a general anesthesiologist should be deft at T&As and BMT-bread and butter cases so to speak.
Now, in response to the above posts, some residency programs have graduated general anesthesiologists who leave feeling 'uncomfortable' with pediatric anesthesia in general. I have an attending from a prominent name program, who scored in the TOP TEN individual scores for the ABA exam, who tells me he had the bare minimum requirement as per the ABA, and now refuses to do any pediatric anesthesia, even with his BC Pedi anesthesiologists nearby.
I will repeat again, for healthy peds, bread and butter cases, you do not need a fellowship, unless you graduate feeling uncomfortable with kids.