Does learning have deadlines? Philosophy vs. Policy

I am reminded today, and for the past couple weeks, why teaching classes is so important to my full time gig as an instructional designer/technologist. Being "in the trenches," so-to-speak has taught me a lot about what theories apply to the real world. For example, I have always been a very liberal-minded teacher. I believe that learning does not have a timeframe, nor deadlines. I often question why we have late work policies and guidelines in our classes. Why would I NOT accept a late paper, if in fact it did prove that the student learned....perhaps a few days or weeks later than all the rest :). If my philosophy is that all students can learn and it's my job to facilitate that, then doesn't my policy of "no late work" counter that philosophy???? I'm really struggling with the insanity that accepting late work brings, because, in fact, I have a deadline to submit grades to the institution. Ugh! So what do I do? Keep my sanity; adjust my philosophy; create a policy that's a bit flexible. However, as for every semester, I have a student that wants to argue my policy about late work. Even when I've allowed them to submit it "due to special circumstances" (my "out" in my syllabus late work policy). They want to push my "late deadlines" to the extreme. "Mrs. Ballard, can I submit all 4 paper assignments two days prior to grades being due?" No! This is where I draw the line. Philosophically, I believe these students can learn, and will learn, but perhaps they need to fail (receive an F on their transcript) for this semester in order to continue on learning the next semester.