When did Evangelicals give themselves the right to decide who's Christian?

Seems like I get into this discussion over and over again. I'm an atheist, but the majority of my family is Catholic, and they definitely consider themselves Christians. But so many Evangelicals I meet refuse to acknowledge that they are - they say they're not "real" Christians, and that they follow false doctrines/worship false gods etc. I've noticed the same thing with Mormons and Jehovah's Witnesses - Evangelicals insist they aren't "real" Christians, even though Mormons and Witnesses definitely do think of themselves as Christians. So when did Evangelicals decide they have the exclusive right to decide who is and isn't Christian?