One of us has a hard-and-fast rule for her take-home exams: turn the exam in within 72 hours of checking it out, or face severe consequences, such as flunking the course. The emergency "out" to this rule is that, if the student sees that he or she isn't going to make the deadline--for a good (read: sudden-onset emergency) reason, that student is supposed to contact someone in the administration to explain the problem and get an extension.
The practice in such a case--when the emergency is real, as opposed to just a failure to follow the rules--is for the student to file a Motion to Accept Late Filing, just as the student would have to do in a real-life legal emergency.
Some students, over the years, have learned their lesson well: some have done a full-blown Motion, and that motion worked. At least one has simply filed a rambling apology that never answered the question of why the student didn't just pick up the phone and call the Registrar to explain the situation, as the exam rules provided. We fear for the future clients of that student.
Why be such a stickler about the consequences for missing deadlines? Because, in real life, lawyers who miss deadlines are committing malpractice, and their mistakes may well prejudice their clients' rights.
Too many law students see law school as an extension of their undergraduate education. But law school is professional school. The habits learned in law school will carry over into the practice of law.
We wouldn't want engineering students to say, "OK, the calculations are off, but our design is close enough that we'll sign off on it, even though the design won't work." We wouldn't want medical students to say, "Well, we know what most of the organs in the body are called and what they do; that's close enough."
Lawyers take people's lives in their hands: sometimes literally (death penalty cases), sometimes figuratively. That's important work. Deadlines matter: see here.
Lesson: If you're going to law school to learn how to be a lawyer, start behaving like one while you're in school.