Early in the novel Holden meets his history teacher Mr Spencer to discuss his very poor exam paper and why he 'flunked' history. During this meeting we learn that not only will he be leaving his current school Pencey due to failing all but one subject, but that he has dropped out of two previous schools,
"If I'm not mistaken, I believe you also had some difficulty at the Whooton School and at Elkton Hills." He didn't say it just sarcastic, but sort of nasty, too. (p. 17)
We could assume that both those schools were also both elite private schools. We learn more about Holden's character at this point as he insists that he didn't flunk at his last school, Elkton Hills, and then muses to himself that he left because he was "surrounded by phonies" (p.14) - a motif which is repeated often throughout the novel.
Holden Caulfield was previously attending The Whooton School and Elkton Hills. He is currently attending Pency, and he is likely to be dismissed from that school as well. Holden is going to be dismissed from Pency for failing all but one subject. Mr. Spencer seems to think that is why Holden was dismissed from the other two school as well, but Holden indicates that he left Elkton for social reasons. Holden tells his readers that Elkton was full of phonies, and that is why he left the school.
One of the biggest reasons I left Elkton Hills was because I was surrounded by phonies. That's all. They were coming in the goddam window.
Holden's obsession with seeing phonies everywhere is a consistent motif throughout the novel. He sees people as fake and consequently puts emotional distance between himself and everybody else. Even people that genuinely want to help him succeed, he pushes away. The only person that he doesn't push away is his sister Phoebe.
The previous answer correctly states the names of the schools that Holden leaves, for one reason or another. At Pencey, his latest school, it seems like he simply didn't apply himself and so he's been kicked out.
However, as a general thing it seems that Holden struggles less with the academic side of things in schools than with the social. He feels like he's surrounded by 'phonies' all the time, both among teachers and among his peers: insincere types, boring types, self-satisfied types, and so on. He attempts to explain his departure from all those schools later to his kid sister Phoebe, although he thinks that maybe she can't understand. However, it is enough that she listens:
But she was listening, at least. If someone at least listens, it's not too bad.
Of course, Holden's complaint about his schools being full of phonies is maybe only a cover for his own inadequacies. He certainly does seem to struggle a good deal with making connections with his peers and authority figures, even taking into account the fact that he is in the throes of adolescence. The only people he seems able to get on with are his own family, at least his siblings: he remembers his dead brother Allie with excessive fondness, and gets on well with his little sister, Phoebe, and also obviously respects his older brother, D.B.