When Morgan Neville began making documentaries 25 years ago, it was almost impossible to get funding or distribution for his movies.

Back then “there was absolutely nothing cool” about the genre, said Mr. Neville, who directed this summer’s surprise hit “Won’t You Be My Neighbor?” The film about Fred Rogers, host of the children’s-television show “Mister Rogers’ Neighborhood,” has made more than $22 million at the box office.

The...