Yes, I remember seeing most of a British movie about this subject. The locals wanted their hill to be designated a mountain, and were trying to influence the surveyors, while also carrying dirt to the top to make it higher. I forget the necessary height. That was in the UK (Eire? Scotland? Wales? can't remember) and I suspect it may vary in other places. In any case, the distinction only matters as a factor in local prestige. If you can't tell by a quick glance, it's a hill IMO.
I live on the US gulf coast, so anything more than 3 feet high is a hill, and mountains simply don't exist. It's a little different in the Rockies.
Speaking of movies, there was a short-lived TV series here that was supposedly set in Houston, TX, and the opening sequence always showed the city skyline with definite mountains in the background. We always laughed, and the rest of the program was ruined for us. It didn't last a full season, IIRC.