Not just the big ones, but the medium and small ones too. I have yet to meet a director of a smaller festival who doesn't believe their event 'may not be in the top 10 worldwide, but is certainly in the top 20'. That list runs into hundreds. Third, at some time in its life, any European film festival will be claimed (not necessarily by those who run it) to be 'the Sundance of Europe'. Film-makers emerging from this cradle will often be referred to as 'the new Tarantino'.
It is in the nature of organisations to become convinced of their own significance and infallibility. When it comes to festivals, what both critics and acquisitions executives (albeit in different ways) crave from festivals is just what they ought to crave: films which fly too low for the radar but which, with the right handling, could soar.
So did the 2008 summer circuit deliver' Let's start with the Transylvania International Film Festival in Cluj, Romania, an event with a hugely enthusiastic student audience who will turn out morning, noon and night for challenging films.
The two Cluj discoveries came, inevitably, in the Romanian programme, starting with Elevator, a first film by George Dorobantu, about two teenagers trapped in a lift. Dorobantu didn't go to film school: he is a riverboat pilot who watches DVDs while waiting for the next ship to show up.
They must be good ones, because you can't tear your eyes away from Elevator: tense, funny, moving, brilliantly edited - and all shot on one set for just $885 (EUR650). Nor is it an 'art film'; Dorobantu is a natural storyteller with an impressive command of his medium.
Equally impressive was Bridge Of Flowers (Podul De Flori), a documentary about a couple of seasons in the life of a Moldovan farming family. Neither stressing nor denying their presence in the room, director Thomas Ciulei and his crew take you inside the hopes and fears of his subjects as they cling on just this side of poverty.
The film is reminiscent of In Spring One Plants Alone which launched Vincent Ward - another film which made being there at the right time seem effortless.
In the UK, Edinburgh is now settled into its mid-June slot and officially devoted to - that word again - discoveries. So soon after Cannes, most of these tended to be among the UK titles, with genre proving the key to a couple of the more interesting entries: thriller Donkey Punch, the latest in the Warp X/Optimum Releasing package which proved both knowing and innovative and which has already opened in the UK, grossing $253,000 (£145,000) to date; and horror film Mum & Dad, which has a controversial business plan (it opens simultaneously across several platforms in the UK on December 26).
For me, the revelation of Edinburgh was a gentler, defiantly non-genre movie: Matthew Thompson's debut feature, Dummy, the story of two teenage boys, one in his early teens, the other 18, who lose their mother to an overdose and stay on alone in their house through one summer.
Treating Gothic material in a matter-of-fact way and with a clever use of flashback, Thompson's film suggests he may be what some have waited three decades for - the new Nic Roeg.
In Toronto, meanwhile, I was on the Discoveries section of the Fipresci jury. One film stood out: Lymelife, directed by Derick Martini and co-written with his brother Steven, which won the Fipresci award. Set on Long Island in the 1970s, the film starts off on familiar territory - dysfunctional families, coming-of-age teens - but slowly establishes a unique world view and an unforgettable collection of characters.
Shot on a shoestring with an impressive cast - Kieran and Rory Culkin, Alec Baldwin, Timothy Hutton, Cynthia Nixon - and with Martin Scorsese as executive producer, Lymelife is emotionally powerful, funny and disturbing all at the same time. What impresses most is the Martinis' refusal to idealise the basically OK young characters or to make judgments about the basically not-OK older ones. I look forward to seeing it again.
Finally, just as the leaves turned brown, I went to my annual gig at the Netherlands Production Platform in Utrecht - still, after 25 years, a gem of a festival - and witnessed the presentation of a fascinating UK-Netherlands-Iraq project called Um-Hussein, directed by Mohamed Al-Daradji, put together by UK-based Human Film and with the highly experienced Antonia Bird and Pippa Cross executive producing.
The story of an old woman and her grandson searching for a soldier who disappeared under Saddam, it begins shooting in Northern and Southern Iraq this month with permission from the Iraqi government but still under perilous conditions (Al-Daradji was kidnapped and tortured while shooting his previous film, the documentary War, Love, God & Madness, a couple of years ago). Um-Hussein is a story of hope dedicated to the great beauty of a country now associated only with explosions and destruction. Seeing that should be a discovery, too.