It seems that some maintain the existence of a "planet X" is all but science fiction, while others claim that its existence is a possibility, even if that's a relatively remote one. Others take this further and make the argument that its existence is likely. Is there a compelling case that such a planet could exist, and if so how come it hasn't yet been discovered - is it a simple case of it being too far away, or is there more to it?
This topic is actually pretty nicely covered in the Planets beyond Neptune Wikipedia page, so I recommend reading it, if for nothing else then for a convenient collection of references. But to quote the most relevant part on Harrington's Planet X (beyond Pluto) of that Wiki page:
Planet X disproved
Harrington died in January 1993, without having found Planet X. Six months before, E. Myles Standish had used data from Voyager 2's 1989 flyby of Neptune, which had revised the planet's total mass downward by 0.5% — an amount comparable to the mass of Mars — to recalculate its gravitational effect on Uranus. When Neptune's newly determined mass was used in the Jet Propulsion Laboratory Developmental Ephemeris (JPL DE), the supposed discrepancies in the Uranian orbit, and with them the need for a Planet X, vanished. There are no discrepancies in the trajectories of any space probes such as Pioneer 10, Pioneer 11, Voyager 1 and Voyager 2 that can be attributed to the gravitational pull of a large undiscovered object in the outer Solar System. Today, most astronomers agree that Planet X, as Lowell defined it, does not exist.
No. A planet of significant mass would cause detectable perturbations in the orbits of the other planets.
This is how Neptune's existence was postulated. Deviations from predictions in Uranus' orbit suggested where an 8th planet must be located. A focused search in the projected area led to its discovery.
From basic physics if there is one it cannot be massive, or must be incredibly far away, as we would see the perturbations in the orbits of the other planets.
Additionally, if it was massive but very low density, for some reason, so we couldn't see it's effect on the orbits of the other planets, we would expect to see something with a regular orbit from astronomical observations.
From what we know of planetary distribution, massive planets are expected, well, about where Jupiter and Saturn sit, with smaller ones closer to the sun, and also smaller ones further out, with Uranus and Neptune the last of the gas giants.