I don't really agree with the concept of getting rejected by 'mistake', as if Oxbridge selection is a mechanical matter of selecting The Best from the Very Nearly Best. There are no absolutes when it comes to assessing the potential of young adult applicants, nearly all of whom have excellent academic records. I was rejected by Oxbridge one year, accepted the second, and went on to get a 1st and do postgrad study.
I don't feel like the college which rejected me made a mistake, and somehow failed to see my true potential. I didn't show them much potential. I was insecure about my A level predictions, panicked by the interview, intimidated by other applicants. Even if I'd received an offer, I think the first year would have been difficult for me.
The second time was completely different: I was confident, with good A level result, and excited to be staying at college meeting people, and my personal statement was full of books I loved rather than books I thought I should admire. I was enthusiastic in my interview, but I didn't obsess about the process half so much, because I was due to go to Australia a week after and was happily distracted by that.
People change a lot between the ages of 17/18/19. Some people also make better applications outside the pressures of sixth form -- I know I got a lot of well-meaning but unhelpful or contradictory advice, first-time around, and the sense of scrutinity from teachers and peers was also hard (who's got an interview? what exactly did you say? have you heard?).
I would encourage anyone to reapply, if, on results day this summer, they see their grades and think: 'I really didn't represent myself well at interview; I know I can do better than that'.
I would be less enthusiastic about 'huh: look, I got great grades, and someone with worse grades got a place. That's not fair. I want another shot'.
I did meet a few people at my second year of interviews who were also on their second shot, and who seemed fuelled by resentment or entitlement, like there had clearly been a mistake made first time. None of them got offers at my college, and I have no idea if they got in anywhere else, but they were not especially pleasant to be around. (And they played a vicious game of Scrabble.)