I worked it out with the professor and I got docked 10% on every assignment, but he also added a bonus question for everyone and that let me stay in the A range regardless.
At least in my admittedly anecodtal experience, most professors are willing to work with you if you are making an effort in the class and have a genuine reason besides laziness.
The lower you go on the academic totem pole, the less tolerance there is. Extensions are hard to get at state schools-- even the really good ones-- and practically nonexistent at community colleges.
It may be unfair, but it makes sense. Teacher:student ratios are lower at elite colleges, and late work is rarer, so it's not really a burden. Also, professors at elite colleges are generally happier and have cushier lives, whereas teachers at CCs are usually underpaid and overworked already, so late work is an intolerable addition. Finally, many of the students in middling schools shouldn't be in college at all, while the assumption at an elite college is that the person who's late is a good student with a good reason. That assumption's not always true, of course, but elite colleges would rather make one type of error.
At private schools, tuition is much higher, parents are wealthier, and alumni contributions are important. So the utility function of the professor is actually linked to the grade of the student (also why you see more grade inflation at the more expensive schools, se, e.g., - http://en.wikipedia.org/wiki/Grade_inflation#Princeton_Unive...).
In state schools, class sizes run much larger; granting an exception with a 200 person class is not likely to remain a secret for long. This means the penalty to professors for not following tit-for-tat is much greater.
Also, professors at state schools just don't have the time available to even consider the excuses - it's easier to consider 4 unique cases (out of 20 students) at a private school vs. 40 unique cases (out of 200) at a public school.
They hold that belief because it is true.