You mean they passed quals but couldn't get an advisor?

Or failed quals?

Are those different things? At departments where quals have high failure rates, it's really more of an annual layoff than anything else.

In many programs, the department aims to admit far more people than will pass the quals. They need the Calculus and Pre-calculus TAs but do not have the advising capacity.

Even if everyone gets a 95% on the quals, the majority will "fail" by necessity because the department simply does not have the advising capacity for the number of TAs they need. Of course, the department typically designs the quals to these needs either explicitly or implicitly.

This is usually at least implicitly understood by the faculty, who will navigate it when absolutely necessary. For example, I've seen it happen that if a professor really needs a student and vouches for/protects them (eg because the research is computational and the student came from 5 years at Google), then the student gets more goes at the plate on quals than is typical.

Depends on the school. My program requires you have an advisor before our version of quals. Our exam was also a research presentation followed by an oral exam where you had the pleasure of doing often a painful amount of algebra on a blackboard. It was also the culture that, while your advisor could ask you questions, they rarely would unless they were helpful (maybe a softball confidence booster or simple version of another question to get you started down the right track) or leading questions for you to talk about something not in your talk. People were largely concerned about being failed by the other examiners, not their advisor at that stage.

However, we were a small department and your acceptance was predicated on at least one faculty member wanting you to join their lab. Some people moved around within their first year, but the majority stayed with the first group they joined.

er, right, what other people call quals we called prelims. our quals were a presentation of early stage work, so when the deadlines rolled around, they didn't have an advisor, or hadn't been working with them long enough to have any results.

It's funny, my CS program in US you had to have an advisor by end of your first year and your quals were basically the thesis proposal you did with your advisor.