I have mixed feelings on accepting late assignments. As noted, we should be treating homework and projects as assignments given to the student by the future employer. If your boss gives you a hard deadline for a project, they are not going to accept it 5 days past the due date. However, collecting these assignments, can give the instructor an assessment of the student's learning and therefore are extremely important. This is why I feel it is essential to establish your expectations on Day 1 and stick with this policy. Also, to set the example, the instructor must also return assignments, exams and other assessment methods to the students in a timely manner, after all, we set the example of the expectation.