Doable to some extent, but would they really learn much that we haven't already told them, given our propensity to Google for symptoms and diseases?
Personally, my worry here, if I had some embarrassing medical history that I wouldn't want people to know about, would be some malicious party gaining access to de-anonymized data and using it to blackmail, or just simply making it public.
Edit: Come to think of it, insurance companies could probably have a field day with a data set like this.