Not at all. I grew up in a household where both parents watched old TV and movies, and I became one of them. By the time I was ten years old in 1977, I knew Bogart and Bacall, Red Skelton, Jack Benny, The Lone Ranger, Robby the Robot, Quasimodo, Sidney Carton, Robin and Marion, Ben Hur, Spartacus, and a zillion others that my peers had never heard of. One of my dad's favorite westerns is Shane, and it became one of my favorite westerns, despite the fact that it was made nearly 25 years before I was born.
So the statement doesn't date me for age, I think, so much as for age of interest. Modern movies are very good. But old movies are great!