The Lazy Student, The Grieving Student, The Sports Fan: make the deadline for reports a curve instead of a cliff. Each day of delay costs some percentage of the grade.

I've always liked the "drop the n lowest scores" strategy. For example, 10 assignments given with the lowest 2 scores ignored.

You are pre-committing to a set of rules, where any excuse would have a much lower probability of being true. Any excuse would need to include 3 excuses. Combining the probabilities of each of the excuses will likely bring the total under your acceptable threshold. Basically, it's lowering the likelihood that you will want to violate the rules.

You can also look at like this. Your model of people predicts that they are scoundrels, and will try to violate the rules, maximizing their utility at your expense. So build a system where procrastinators can maximize their utility at no expense to you.