When I began my teaching career at the high school level (mostly 9th and 10th graders), I had a no tolerance policy. I didn't take anything late.
I still had a no tolerance policy. Get your work here on time, or it doesn't count. Sorry.
And then we did a faculty book study. I believe the book was Whatever it Takes by Rebecca DeFour et al.
The book discussed 0's and how they let both the student and the teacher off the hook. The student didn't have to do or learn the material, the teacher didn't have to ensure the student did or learned the material.
I realized not accepting late work and handing out 0's was taking the easy way out.
So I began to take late work but with points deducted for every day the assignment was late (I don't generally give any grade lower than a 50, though, as long as I receive the assignment).
And I kept this policy for the next several years. It worked pretty well. Students finished work, I encouraged them to finish because they could still get credit, and students were held accountable for not having work finished when it was due. I was happy, students were happy, parents were happy.
And then this past summer, I discovered Rick Wormeli. I shifted once again. This school year I implemented a broad retake and redo policy (detailed in this blog post). And I began accepting late work with no penalty. The idea being that if students demonstrate they know material - even late - the grade they receive should reflect what they know, not their behavior.
It's working well. Students are willing to finish work and give it their best effort because they know they're receiving full credit. There's no falling of their demeanor by learning they can do the work but only receive a few points.
It's really not any more work for me than my previous turn-it-in-late-but-receive-points-off policy.
And my students' grades are a more accurate reflection of what they actually know.