Is it possible for a PhD applicant to be rejected against the potential supervisor's will?
Yes, it's possible.

Practices vary, but in most US universities, graduate admissions are not decided by individual professors, but rather by a committee of faculty.  Their decisions often must also be approved by a graduate dean or another higher-level committee.

There are many possible reasons why an applicant could be rejected, even if they have the support of a potential supervisor:


The committee might disagree with the supervisor's assessment of your qualifications.
The professor might have a record of questionable judgment in recommending PhD students.
Many PhD programs include coursework and exam components before beginning to work on dissertation research with an advisor.  It could be that the committee isn't confident that you would pass those components.
The program, or the university, might have strict minimum GPA or GRE score requirements, that you might not have met.
The committee might not feel that the professor would be a good supervisor.  This could happen if he does not have a successful record with previous students, or if they feel he already has too many students to advise well.
The program might have a limited capacity for students, or a limited amount of funding.  Even if the committee agrees with the professor that you are well qualified, there might still be more qualified candidates than they can accept.  Maybe there are even too many candidates who have the support of particular professors. If so, some good candidates have to be rejected.  This is probably the most likely reason.
It could be, as the professor seems to think, that a mistake has been made, and his recommendation has been unintentionally overlooked.  If that is so, it is possible, though not certain, that the decision could be changed.