It happened again this year. At Cambridge, it's true, slightly more candidates were accepted from the maintained than the independent sector - 47 per cent against 45 per cent - and the figure for the maintained sector was the third highest proportion since records began.
But that, as Susan Stobbs, chairman of the university's admissions forum, remarked, still represents only "a slight lead" by the maintained sector. And at Oxford University, the position is worse. This year, the share of successful applicants from the state sector fell from 43.6 to 41.6 of the total, lower than at any time in recent years.
Two years after Oxford abandoned its special entrance exam, which was thought to favour independent-school candidates, those candidates romped home with 49 per cent of the total 3,200 places.
Every year, bewildered parents of disappointed applicants from state schools with a string of A grades write to the newspapers to complain of unfairness. But three As are nothing exceptional to the admissions tutors at Oxford and Cambridge.
Most successful applicants to the two universities (and as many as half of those rejected) go on to get three As. The usual request is a minimum of two As and a B at A-level, or 28 points. This year, that level was achieved by more than 85 per cent of the successful applicants to both universities.
Although independent schools have only 7 per cent of the school population overall, they account for about one-fifth of pupils staying on in school sixth forms. And the schools are very good at A-level teaching. Nearly one-third of the applicants getting into university with 26 to 30 points (ABB to AAA) come from the independent sector.
Provisional 1997 figures from the Southern Examining Group, which assesses almost a third of all A-level exams, confirm that pupils from independent schools are far more likely than others to get A grades.
In English, for instance, more than one-fifth of independent-school candidates got an A, compared with less than one in 10 of all candidates. In biology, 30 per cent of pupils from independent schools got the top grade, against less than 15 per cent overall.
But even allowing for the high proportion of the best A-level candidates in the independent sector, Oxford and Cambridge still have quite a bit of ground to cover.
One of the chief problems, according to both universities, is getting good candidates from the state sector to apply. Both have made determined efforts in recent years to try to attract more and to dispel, in the words of an Oxford spokesman, "ill-founded and old-fashioned misconceptions of social litism".
Interviews, a vital part of the admissions process at both Oxford and Cambridge, are another serious hurdle for many applicants from the state sector.
Many pupils are simply not used to defending their point of view in a one-to-one interview, and their schools do not have the resources to train them.
So they may not be able to convey what it is that Cambridge says it really wants - not a string of As but "a desire to study almost for its own sake, an excitement in their subject and in our courses".