I am on the fence but leaning more towards the "he's for real" side as more and more of the predictions come true.
There are a few things that make me think he must be fake though, like for example, someone on anomalies brought up the question of "where is the time machine/car?" -- you would still see it at this time while it is on it's journey to the future. Well maybe not since he had to go to 1998 first but you would think there would probably be another that ended up in this worldline on his trip from 1975 to 2036. Maybe the trip is made inside his parents garage but that then makes the garage unusable? And how do they keep 5 year old JT from ever seeing it? Kind of like how Clark Kent never saw his spaceship until he was a teenager in Smallville? What if it crosses a worldline in this garage where no instance of Titor from another worldline ended up stopping there? What are they going to think of this anomalie that appears to form there?
They can't even hide it in the basement because there basically are no basements in FL because the elevation is too and the ground is basically a floating chunk of limestone & dirt. Which creates yet another problem with the dirt underneath the car moving with him through time as he explained it. The hole left underneath the car would likely hit water, and the time machine along with the ground being displaced could sink.
The trip would have to be made out in the boonies somewhere that isn't even developed in 2036 (or in the past). It should show up on satelite photos....
Despite these things which seem to make it impossible, with all the predictions coming true perhaps we are just not thinking far enough "outside the box".