A good friend of mine has been teaching piano for over 20 years and this is a persistent issue with her students. The first thing she has to teach them is how to get over making mistakes. Once the kids get past the point where they're no longer progessing rapidly, they get frustrated and want to quit. There's a point in learning most skills where the going gets slower,you have to work harder and you make more mistakes. Getting over that hump is the key to success in anything.
She talked to one student about beating a video game and how many times he "died" before he got to the end. Kids who will spend hours getting clobbered in a video game don't seem to be able to translate that into other things. Once this student got that his piano lessons were made up of levels like his video game, and as you got to the higher levels the "battles" got tougher, he was able to stick with it.