Percival Lowell, who is most well known as a proponent for canals on Mars called this hypothetical planet Planet X (X for unknown)! He performed two searches for it, without success, the first ending in 1909, and the second started in 1913 after revising his prediction for where it should be found. This search ended in 1915 after which Lowell published his theoretical results parameters for Planet X. Ironically, at his observatory that very same year, two faint images of Pluto were recorded, but not recognized as a planet until after the discovery of Pluto in 1930.
Pluto was originally thought to be Planet X, but Pluto's mass was not sufficient to explain Neptune's orbit, so the search continued. However these apparent discrepancies were resolved when the Voyager 2 space probe discovered that Neptune's mass had been badly miscalculated; with Neptune's newly discovered mass taken into account, there was no longer a need for any new planet to explain Neptune's orbit.
Our most powerful detection techniques are capable of detecting an Earth-sized planet 70AU from the Sun, a Uranus-sized planet 90AU away, and a Jupiter-sized planet up to 120AU away (neglecting its gravitational effects on the Sun). Of course, the sky is very big and the most powerful telescopes can only look at a very tiny fraction of it at a time. Pluto, for comparison, is around 45AU away at the moment.
If a tenth planet exists, it is unlikely to be native to the solar system: comprehensive surveys of the ecliptic have been undertaken, concluding that no planet of Earth size or greater exists in the ecliptic plane closer than 60AU. Thus, any tenth planet would have to be in a highly inclined orbit, and so likely to be a captured object and not one that was formed with the solar system.
wikipedia.org dumped 2003-03-17 with terodump