This has happened at least once a semester since I took this job. A student who has been underperforming all semester--turning in half-assed homework assignments, missing a lot of class, earning failing grades on exams--realizes suddenly that he or she is going to fail the class. But it's after the late withdrawal deadline, so there's no simple way to get out of it. So they write me an email or come to my office and ask me to give them permission to obtain a late withdrawal.
Now, it's really not in my power to simply grant the late withdrawal. It is in my power to let the dean know that, in my view, the student has a legitimate reason for the withdrawal, and unless I do this, the dean will not normally consider the student's request. But it's not as though the dean acts on my say-so. It's the dean's prerogative, and I can't imagine that my input would count for much.
I am nevertheless asked to grant late withdrawals all the time, and since it's university policy to grant late withdrawals only in cases of extreme extenuating circumstances (or it ought to be), I always ask what the extenuating circumstances are. In several cases, the extenuating circumstances have been, "I will fail the class unless you let me withdraw." These student thereby demonstrate incomprehension of the meaning of the expression 'extenuating circumstances.' One time the student told me that since the university has already taken her money, the F on her transcript would be a "double punishment." I mean, where to begin?
My guess is that most of this stems from a pervasive It Doesn't Hurt To Ask mindset. But in typical cases, the student is persistent--making several requests over the course of days or weeks. And I've had a couple of students send me borderline belligerent emails, or go over my head and complain to my chair about this. I like my job a lot, and although I have been compelled by these experiences to reflect upon the justification for assigning grades and holding students responsible for their academic performance, overall I find this aspect of it really, really annoying.