Is The Fall movie based on real events?

The movie is not based on real events, but is a retelling of the 1981 Bulgarian film Yo Ho Ho (which itself is an original story).

The Fall is incredibly realistic feeling at times, one of the reasons of which can be derived from a quick look at its Wikipedia Page:

[The director's audio commentary] reveal that actor Lee Pace remained in a bed for most of the early filming at the director's suggestion, convincing most of the crew that he was in fact unable to walk. The intention, Tarsem and Pace noted, was to maximize the realism of Roy's physical limitations in the eyes of Catinca Untaru, whose lines and reactions as the character Alexandria were largely unscripted, and so were young Catinca's spontaneous interactions with Pace's character. For example, Alexandria's misinterpreting the letter E as the number 3 in a note written by Roy came about from an accidental misreading by the 6-year-old actress during filming, which the director then realized he could adapt into a clever twist in the story.

To further the realism of young Catinca's performance, Tarsem had portions of the hospital scenes between Pace and his young co-star filmed through small holes in the hospital bed curtains, maximizing the youngster's spontaneous interactions with Pace despite the presence of the film crew surrounding them.

I feel this, coupled with the shots towards the end of the film of some of the most dangerous stunts in silent films, make it seem like it was a real story - but alas, there doesn't appear to be any evidence to suggest it is.