As per Rick Steves' book, I called 2 months ahead for tickets to the Last Supper for the end of June. Much to my dismay, all the slots were completely full. I asked about cancellations and they said that they would only make them available on the Saturday, one week before. So we got up at 4am to call 1 week before and they had numerous slots available to choose from!
So if you are running into the same issue with not being able to get tickets, I recommend asking when they will post the cancellations and then call back at that time.