It depends on a lot of different factors.
Sometimes life just gets in the way, as in the case of people who become seriously ill (I know people who have dropped out or taken time off due to clinical depression), get pregnant, or have cash-flow problems. While obviously a reduction in school costs would decrease these instances, it wouldn't completely eliminate them.
Some people get to college and then decide that they're happier doing something else. That isn't somebody's "fault," it's just another kind of education. Bill Gates is one famous example of a college dropout who isn't complaining about it.
Then there are the instances where it is the fault of the school. My alma mater, George Washington University, is notorious in the academic world for not doing a good job of letting students know about graduation requirements, deadlines, policy changes, etc. The advising system does not support the size of the student body, and this results in low graduation rates. The correlation here between the lower-than-average number of students who graduate in four years or less, and the recurring problems with the advising system, suggests to me that it is the school's fault.
And then, there are some students who just have no idea what it takes to be in college. When my fiance taught in a community college, he had students come to him in his office and say "hi, I've missed 17 out of the 20 class sessions, but now that the final is in a week I really want to pass the class. What can I do?" People like that tend to drop out because they're not really invested in the first place. Even a great school can't do much with someone who doesn't even go to class.
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