Fast-talking Hank Green very concisely explains why there might be at least one planet larger than Earth well beyond the orbit of Neptune (and in passing why most astronomers think that's unlikely):
Follow the link to the YouTube page and click on "More" under the description to see links to the original papers and other articles about the subject.
The video runs under four and a half minutes, but here's an even shorter summary:
All the full-fledged planets orbit the Sun more or less in the same plane as the Earth, but more distant bodies, including Pluto, have orbits that are more tilted. Hence they spend part of the time "above" the plane of the Earth's orbit and part of it "below".
Now here's something odd: a lot of these distant objects cross the plane of the Earth's orbit at just about the same point in the orbit that's closest to the Sun (the "perihelion"). They might just as well cross the plane anywhere else in their orbits, so this is curious coincidence. A team of astronomers has computed that a distant planet about ten times the size of the Earth could explain the phenomenon, and if it's far enough away it's not surprising we haven't seen it yet. On the other hand, our understanding of how the solar system formed makes it unlikely such a massive body could have formed that far out. Meanwhile, another group has looked at a different line of evidence that suggests the existence of a distant planet, but not the same one.
(And no, none of this is evidence for the supposed mystery planet Niburu.)
(Updated for clarity and to correct a typo.)