If you're talking about asking for an extension 10 minutes before the deadline (or 10 minutes after), that's a whole different kettle of fish.
Perhaps this has more to do with the role college plays in American society. I believe that college has become more of a high school extension than anything, and so maybe it is wrong to blame students who treat it as such. I doubt they would be slacking off in grad school (but I can't be sure!)
Given that deadlines in education are more or less entirely arbitrary, I don't see the harm in extending them within reason, and providing students with the opportunity to successfully complete their tasks.
As for tardiness -- it happens. If someone consistently fails to meet our estimates, but delivers a sufficient degree of value within a reasonable amount of time, then I'll simply learn to incorporate their individual turn-around time in my estimates.
Hard deadlines are generally driven by what are ultimately financial/market concerns. I sweat the deadlines when a trade show is approaching, and I sweat the budget when we see schedule slippage on an release, but most of the time I'm happy as long as everyone is productive, progress is being made, and the budget looks OK.