I also have the understanding that most schools are lenient about LORs coming in slightly late. For instance, if the the deadline is a Friday, I really couldn't see there being much adverse reaction to an LOR sent in 5:00AM on the following Monday. No one will have processed the applications by then. I think for several December 15 deadlines, the goal may be to have all applicant information set and ready to go around the 1st of January, so admissions committees can make fellowship decisions by late January. Of course it varies by school.