Few, if any, schools are full. Most, however, are close to full. Historically, Case and Columbia are usually packed by Dec 1st.
Students turning down schools have been used as a means for comforting those who havn't gotten into schools. However, this is to a great extent a means of creating false hope. Yes, many students turn down schools' offers, and the schools anticipate this: they accept many more students than they have spaces for. After Jan 15th, they're just trying to fill up the few remaining spaces.