We're not saying it's impossible, but on the show a lot of the theory rests on the idea that "the dragon has three heads" (claimed by Rhaegar in "A Clash of Kings") — those heads being Jon Snow (confirmed Targaryen), Daenerys (confirmed Targaryen), and Tyrion (unconfirmed Targaryen). And Dany has three dragons that need riders. However, in the books, we already have a third Targaryen, Aegon, who would be Jon's half brother, the son of Rhaegar. He was smuggled out just before the Lannisters took the castle. Westeros widely believed he was killed by their forces on the night of the sack, but Varys switched another baby with him. His existence is confirmed in "A Dance with Dragons," when Tyrion meets him. 

So, the theory isn't impossible, but in the books we already have three confirmed Targaryens to ride the three dragons.