I suspect that this claim is more of an excuse by students who do not want to take the class during the 8am section where there are plenty of spaces, or those who voluntarily take 12 credits instead of 15 or 16 credits per semester because they cannot handle the full 15 or 16 credits, or those who do not pay attention when class registration time approaches, so that they register later than they could have registered and find fewer choices.
A more legitimate issue is that some students’ schedules are constrained by the need to work to earn money to pay for college, so they may not be able to take the classes they need if the available times conflict with their work schedules. This is probably more of an issue at public schools (particularly those in states where in-state financial aid is poor) than at elite privates where half of the students come from top 2-3% income/wealth families, and those on financial aid get generous grants that mean less need to work.
But it is likely that the low graduation rates at many colleges are student-related, rather than being due to anything that can be construed as lack of class space. Examples include needing remedial courses, failing courses, taking lower course loads, etc… Note that many colleges have four year pledge programs like these:
http://undergradstudies.csusb.edu/advising/fygp.html
https://www.cpp.edu/~academic-programs/graduation-pledge/
https://www.fullerton.edu/aac/Current_Students/Finish_in_Four/index.asp
http://www.csub.edu/academicprograms/Undergraduate%20Studies/Four-Year%20Degree%20Pledge/
http://www.fresnostate.edu/academics/facultyaffairs/documents/apm/202.pdf
But the actual four year graduation rates at the above schools are low, indicating that few students fulfill the conditions for the four year pledge programs.