Despite being labeled as a "gifted" child, I am somewhat baffled by the system as it stands now - for many of the reasons the posters above have listed: identifying children in kindergarten, foreign language tutoring for 3 year olds, parents focusing on this as a goal, instead of adapting to their children's educational needs and learning styles, etc.
I grew up in a very AP home with a very crunchy mom. She certainly wasn't "training" us, simply exposing us to all kinds of experiences and nurturing us. I went to an open elementary school (no tests, no grades, no homework, self-directed work) through 3rd grade which totally fostered a lifetime love of learning. When we moved and I switched to a very conservative school, they were doing work I'd done on my own 2 years prior, immediately IQ tested me and put me in the "gifted" program. For my own benefit, I'm very glad that I was simply pulled out for additional classes, etc instead of encouraging the skipping of a grade or two, since I think this would have been very difficult, being even more different than everyone else.
I did high school (and college and MS) in the South of England, where a totally different type of educational system prevails. Though I was not in gifted classes, I was indeed enrolled in a grammar school. Though completely state funded, children are tested in the last year of primary school. Those who pass go to grammar schools. Those who don't go to comprehensive schools.
I guess I see schools using gifted programs as a way to do some streaming of children in order to provide educational opportunities to fit their needs, much in the same way as if I was learning disabled, while staying within the constraints of a very mixed group of children within a single building.