This is a great question! I always worry about this with large public universities, in the face of so many budget constraints these days. One issue that can happen is the lack of getting the first class in a sequence. You could be shut out for a year if you can’t get into the first of a sequential class in the fall. Sometimes a class is only offered every other year and that can be a problem, too.
Some kids in the community colleges have to attend 3 or 4 community colleges to put together a patchwork of classes, which isn’t ideal.
Even at the expensive elite schools, you can’t always get your courses. But just getting a certain professor or time of course that you want is not the same issue.
Also, if you are in the Honors program, you usually get priority registration which can help a lot.
I have heard of students having difficulty getting the classes they need at University of Washington, Michigan, Cal Poly, Oregon, NYU, Boulder, Wisconsin, Illinois, and Arizona.
This is often a matter of budget because colleges cannot afford to offer every class every term for most students to graduate in exactly four years.
Talk to current students about their experiences. Ask how priority for registering for classes is
determined. Some schools do this by grade (Senior, then Junior, etc), some by special status (physical needs, athletics, Honors students, etc).
It’s also good to know, within your major, whether there is priority given within a particular department for students who have declared a major in that department. This is particularly important in music and art, but you should also think of this for very popular majors. Ask, too, about taking classes in other departments, since some colleges allow this and some do not. Great question!