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What would happen if a student was admitted to graduate school (only consider US institutions here), but failed a course later and would have to wait till the next time to take the course to make up for it and get the bachelor degree?
Do they have a chance in this situation to apply for a deferment of admission for a fraction of a year?
I suppose it's okay to ask. But since graduate admission is generally contingent on successful completion of your undergraduate degree, I think it's much more likely that the graduate institution will rescind your admission. You would need to make a fresh application in the next admission cycle, with no guarantee of acceptance, and the failed course would tend to weaken your application.
As I mentioned in the comments, there are a few cases here.
The best case is if you can make up the necessary course before grad school starts. In the US, you might fail a course in the spring but be able to make it up over the summer. Note, you can usually take courses at a different institution -- so if you fail a course in the fall and your institution will not offer it again until next fall, I would try pretty hard to find another institution where you could take the course over the spring, and then transfer the credits. If you can find some way to pass the course before graduate school beginning, then there is unlikely to be an issue.
The second-best case is that the course is completely unrelated to your degree -- e.g., you are a physicist and you failed music and therefore won't get your degree on time. This is bad, because the university could refuse to accept you over this, or could make you defer an entire year. But it is likely that a work-around can be found. For example, you may be able to make up the music course at your new institution and then transfer the credits to your old institution to get the degree.
The worst case is that you failed a major course -- e.g., you are a physicist and failed E&M. As before, they could revoke your admission over this, or force you to wait an entire year. But in this case, they are far more likely to rescind your admission, for obvious reasons.
As Buffy pointed out, deferring your admission for a fraction of a year is difficult, unless the program routinely does winter admissions (and most do not, at least in the fields I'm familiar with).
