It depends on what you mean by Planet X.
Before Pluto was discovered astronomer Percival Lowell made a prediction that there was another planet beyond Neptune. He did this by observing slight deviations in the expected orbit of Neptune. (The same process has allowed astronomers and mathematicians to predict the existence and location of Neptune based on slight irregularities in the orbit of Uranus.)
Lowell looked, but died before anything was found. Clyde Tombaugh resumed the search. By this time the mystery planet was called "Planet X." Eventually Tombaugh did discover a new planet, Pluto, but it was smaller than expected and not where Lowell had predicted. It is likely that Lowell was simply wrong and Tombaugh got lucky...
...sort of. We now know that there are many objects in the far reaches of the solar system (beyond the orbit of Neptune) which are the inner part of the Kuiper belt. One of these has been found to be larger than Pluto.
So, while Pluto (Planet X) retains official planet status, it is largely for historical reasons. Had we discovered Pluto in the 1990s and not 1930s it is likely Pluto would never had been classified as a planet.
However, it is possible that there is a large, say, Neptune-sized, object out there. Some evidence for this can be found in the distribution of Kuiper objects. But it is still very uncertain. Such a distance planet, could be called Planet 10, or, for roman numeral fans, Planet X. Such a large distant object would be very faint and move only slowly across the sky (making it hard to distinguish from a star).
Another meaning for "Planet X" is a Nibiru, a fictional planet based on one person's faulty knowledge of Sumerian artifacts, which has been taken up by various new age and pseudo-science looney groups. A couple of years ago "Nibiru" was supposed to pass close to Earth leading to floods, earthquakes, death, destruction, yadda yadda yadda. This "prediction" being based on one woman's (Nancy Lieder) telepathic communications with various aliens.
Just as an aside, Ms. Lieder's aliens also told her that the Hale-Bopp comet of a few years ago - which lead to the suicide of one band of cult members - was just a "star." To make sure she wasn't swayed by actual observations of the cometary tale and motion across the sky she simply refused to look at it. Those pesky aliens!