The short answer is no. If you had asked this question 10 years ago it would truly be no. But in 2015 more people are getting degrees in film, plain and simple. Millennials are the education generation. Our parents encouraged/enforced education down our throats from the moment we could blink. So unlike the days of old, when high school graduates or hard-working immigrants applied for lower-level jobs in Hollywood that would become networking heaven and eventually turn them into directors, executives, producers, and writers... these days, film school graduates have their own living network.
Film school students are not only being taught by current industry professionals, but they are being placed with mentors upon graduating. Beyond that, they are being hired by fellow alum. I receive a newsletter a few times a week that lists FSU alums who are working on the same sets. If anything, this isn't news to you, Hollywood is about who you know. Many of the people breaking into Hollywood these days don't have nepotism. So they use the next best thing, their Alumni network. Like other industries, film school students know what skills another graduate has simply because they were in the same program. There is a common bond which makes hiring them a lot easier of a decision than hiring a stranger.
Now, don't get me wrong, you still have people showing up to Hollywood at 18 and working their way up. Experience matters far more in that world than education. But beware... as the decades continue to fly by and the generations continue to get educated, I see the industry changing. One day, one of the first questions you'll hear from another Hollywood-type will be, "What film school did you go to?"