Late submissions are definitely a problem Mayuresh. I think you need to be clear that your job is to teach and also assess the learning objectives. If a student does not provide you with a method to assess their mastery of the learning objectives, then late-excuse or not, they cannot get a good grade.
I know you already know this, but I will re-state this at the risk of sounding dumb - When it comes to accepting late submissions, I would go with the strict letter of the rules in the catalog or other student manual for your school. Never give a student preference over another student by skipping the rules. It may look like they "love you" for doing that, but they will put you down behind your back, tell everyone of your favoratism, and anger the hardworking students who have just had the value of their diploma lowered.
