According to Disney CRO, it starts to get busy the Saturday BEFORE Thanksgiving, and gets more crowded up to Thanksgiving Day. The worst days- meaning for me the most crowded- are Wednesday, Thanksgiving, Friday and Saturday. Then, the Sunday after Thanksgiving it dies down quite a bit until the weekend before Christmas. I think "abamick" has their information incorrect.
One year when we flew in on Thanksgiving day, we were told the Magic Kingdom was closed due to the crowd size around noon. We went the next day to Animal Kingdom and it was nuts. By Sunday, the MK was OK, some crowds, but not bad.