common word in name, two different parts

I think I can help a little here. Tonsil is referrring to the appearance of the part. Most of the time when people talk about tonsils, they are the round things at the back of the throat that get inflammed with strep throat because of being made of lymphoid tissus (that fights infection). The cerebellum is the back of the brain important in balance and coordination. These are most commonly referred to a hemispheres, but can be called tonsils because they look like tonsils and lots of medical names were made centuries ago and are descriptive not functional. The ectopic (out of typical place) cerebellar (referring to the cerebellum) tonsils (round pitted appearing parts) can bulge down enough to cause problems like headaches (or most commonly not). There is a continuous from normal to a variant of normal to Arnold Chiari malformation when the bottom of the cerebellum can dip into the opening from the brain to the spinal canal. From what you have said, you can be reassured that her tonsils (in the back of her throat) do not need to be removed because of this illness and one really does not want to remove parts of the brain (!). It sounds from what you said from the report that the radiologist was only being complete and saying anything that was a tiny bit different in Emily. This does not mean it is related to the CIDP, but it does give information if in adulthood she has bad headaches, her doctors could think about if due to changes over time from a tendency to have a Chiari type malformation. Tonsils (in the back of the throat) are “normally” large in children because the immune system is very active in learning to protect against infection. Tonsils do not have to be taken out unless they are huge and interfere with swallowing or breathing in night (usually more due to adenoids in the back of the nose). I do not know why someone got better from CIDP after having their tonsils out, but would think it is true, true, and unrelated like the brain report on Emily.
With Hope for cure of these illnesses (and understanding)