How common is it for a student to finish a Master's degree at a school and to be rejected for a PhD at the same school? This question likely doesn't make much sense in the US and other countries where separate Master's degrees are uncommon. However, here in Canada, the vast majority of graduate students must first apply for a Master's degree, and then apply for a PhD program (usually filling out applications during their Master's degree, as the deadlines fall).
Students are highly discouraged from completing their undergrad, Master's, and PhD all at the same school for many reasons. But it seems quite common to do one's undergrad at one school, then both grad degrees at the same school.
My question is: how common is it for students in this situation to be rejected from a PhD program at the school where they are currently doing their Master's? And what might be reasons for such a rejection? I imagine that some reasons would be lack of willing supervisors (which begs the question of how they were accepted for the Master's anyway), unsatisfactory work during the Master's degree, or some extreme situation such as academic dishonesty.
My field is math, so I guess we can restrict the question to that discipline, but I'm curious about answers from all fields.
One very obvious thing would be availability. My department has around 100 masters students each year, but will only have 7 PhD places. We pretty much take as many masters students as we can (because they pay). As you can imagine, this is not always popular with the people that have to supervise them for their dissertations.
Master degrees are funded by the student (either directly, or via a loan). PhD positions are funded by external funding bodies or scholarships.