Do Course Deadlines Matter?

Have you ever gotten an email that a student sent in the middle of the night before an assignment was due:
Dear Professor,
I really hate to do this, but earlier this evening I received some really distressing news from home and I haven’t been able to concentrate on finishing my paper for your class. I’m so stressed and anxious that I will go to the student counseling center as soon as it opens, but in the meantime, I won’t be able to get my paper to you in time for class. Will it be okay if I turn it in on Wednesday instead?
Dear Professor,
I’ve been throwing up all night and am so dehydrated that I may have to go to the hospital.

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