If you read the rules of splendor, you only get a fixed number of nobles (number of players + 1). But in this video they add a new noble every time one is taken. Did I misinterpreted the rules or do they use "other" rules in the video?
No you do not; they are playing incorrectly in that video.
From the rules:
Shuffle the noble tiles and reveal as many of them as there are players plus one (example: 5 tiles for a 4 player game). The remaining tiles are removed from the game; they will not be used during the game.