‘Want to see how long a minute is?” Fred Rogers once posed that question on “Mister Rogers’ Neighborhood,” and answered it by setting an egg timer, then waiting in silence for the thing to ring. Want to see the passage of time made manifest? Watch “Won’t You Be My Neighbor?”—Morgan Neville’s stirring and provocative documentary feature about the man and his show, which first hit the American airwaves 50 years ago.
On the surface the old episodes—some of them represented by clips from fuzzy black-and-white kinescopes—are...