It never really crossed my mind if there was an upper limit as to the number of students a given professor can take on. This is because some professor have dozens of students, while other have only a handful. I know a professor who has 2 PhD students and that's it. This seems to be a department dependent variable.
However, a rumor got out that one of the professor in my department (not my own) was attempting to take on a student but since the student did not have external scholarship, therefore the professor was not able to take him on.
It is unusual because students do not need external funding to get into the graduate program at my school. I hypothesized that the professor's funding cap was all used up by his current students, therefore he could not take on anymore students without additional funding. However, I am not sure how this funding cap mechanism work.
Is there an upper limit to the number of students a professor can take on? If so, what determines it? An insider's perspective on this would be greatly appreciated.
(My question is mainly with respect to engineering and STEM major within North American context)