We were there the 1st week of June in 2008. Being from South Carolina, the temperatures were about the same as home. You could notice a steady increase in crowds as the week went on but nothing near real summer crowd levels. Schools around here are just starting to let out at the end of that week.
The advantage was Disney had already switched to the later hours. Night extra magic hours were great. Almost no crowd at all especially the later it got. It was also warm enough for the water parks (we usually go in the winter), so that was great.