I have some pretty tough course policies. I don't accept late papers. I don't accept papers more than five minutes late. I don't even accept e-mailed papers. There are reasons for this, developed over the course of a few semesters' worth of teaching.
I've learned that if I don't require students to hand in their papers within the first five minutes, I'll get a good portion of the class wandering in up until the very end of class, all of whom had problems with traffic or printing or their grandmothers dying (for the 12th time) or a medical emergency or their goldfish running away to Mexico.
Or they have car or work or other scheduling issues that make them miss class; and/or they have printing issues and can they e-mail-me-their-paper-as-soon-as-they-get-home-which-is-by-noon (and which is almost invariably hours later than that because of unforeseen circumstances)?
Or I have students who email me their papers either after class has begun, and often enough several hours after the class has ended, because [insert excuse here].
I eliminate all these excuses and tell the students that their papers are due within the first five minutes of class; that I don't accept emailed papers or papers due after the first few minutes of class.
We talk about these policies several times before the first paper is due, and my policies always catches some students unaware. I also reiterate, though, that if there is an actual problem, when is the time to tell me about it? (And a few students will say, "Beforehand.")
So far I've had one student email me her paper at 8:33 a.m. She'd worked late and didn't get up in time, lives an hour away, and hoped I would accept her paper. I said that unfortunately I could not accept her paper via email; it was against course policy. And despite a follow-up request, I couldn't accept her paper should she arrive by the end of class, nor do I offer extra credit. Two students meandered in 10 minutes late, and one student didn't show up at all, although to be fair he did email me late last night to ask a question about citations. (He consistently arrives 30-45 minutes late.)
At our last class, I assigned the next big writing project; today we're meeting in computer labs with a librarian, who would talk about databases and research methods. By having my students' first paper due today, I guaranteed that most of them would show up with their papers - on time. It was always a problem getting even most of the regularly attending students to show up.