I can't believe no one has mentioned the one that comes to my mind first when I think of songwriting movies.
Thing Called Love
(or maybe it's AThing Called Love

It's set in Nashville... a lot of it at the Bluebird. Shortly after watching the movie I met Barbara Cloyd who runs the open mic that the folks in the movie perform at. Was surprising how close they came to the actual events. Some differences, of course.. but not near as many as I expected.

There's one other I think of... that hasn't been mentioned... (had to go look it up) Grace of My Heart. The main character in that was a Carol King sort. Might even have been losely based on her. Follows this gal's rise from writing in New York for girl groups... up through her eventual solo act.