As someone who has also been involved in Oxbridge admissions:
(1) There is intense competition in most subjects. Inevitably, many very good people will not get places. And, though, great efforts are made to select the very best, inevitably luck will play some part.
(2) Most people apply before their A level results are known. Predicted grades are a significant part of the criteria for acceptance at most universities. And this means both that some people are accepted but 'miss their offers', and that some don't get accepted because they're predicted to get less good grades than they eventually do.
(3) Oxbridge is definitely NOT looking for poshness, or seeking to weed out state school applicants. It is true that private school applicants are often advantaged by better preparation for exams and interviews, and teachers who know the system better. There are increasing attempts to take these factors into account- perhaps not sufficiently, but lecturers are definitely not looking for posh students; if anything, if two applicants perform equally well, they are likely to prefer the one who had fewer advantages and less coaching.
(4) There is sometimes a misunderstanding of the interview process. Interviews involve assessing the student's ability to think clearly about their subject; to answer questions relating to their subject, which do not involve previously-learned facts so much as the ability to solve problems. Interviewers may also be seeking to assess interest and motivation for the particular subject. However, the interviewers are only interested in applicants' academic potential in a particular subject. Interviews are not for the purpose of assessing personality, or 'all-rounded-ness', or ability to 'fit in', or extracurricular activities (which are not taken into account). Contrary to what pp have suggested, interviewers are not looking for confidence, and are only looking for extensive knowledge about the world at large if it's relevant to the specific subject: i.e. it may be relevant to politics or geography, but not to mathematics or classics.
(5) Many subjects set an aptitude test (e.g. the TSA for psychology; the BMAT for medicine and biomedical sciences; the MAT for mathematics). If an applicant with good predicted or actual A level grades is not accepted, one possible explanation could be a less than outstanding aptitude test score.
(6) Luck does play a part!