New Year's Day was crazy last year. Just to give you an example of the crowds... It took 40-45 minutes to WALK from the castle to Tomorrow Land.
That walkway is wide, but when you have thousands of people going in both directions... some with strollers... it can get very crowded.
The parks opened up back area walk ways to help with the congestion, but still, I have never seen the park that crowded!
In terms of the parks closing due to capacity, it happened several times last year. Annual Passholders and Resort guests are not affected by this as previously mentioned. Those with Executive passes to the parks can also get in when the parks are said to be "closed" due to capacity!