I have a hard time believing this. I have had a bit of contact with both Gypsies and Travellers. Their languages have distinct similarities. The pictures on the link you provided show horse drawn wagons just like the Gypsy wagons of generations ago. Both Gypsies and Travellers also used to use tents with an almost identical hoop configuration as the wagons they used.
IMO, there is ample evidence that they have a common background that split many generations ago.
Similar wagons and borrowed words would definitely indicate that Travellers and Gypsies/Roma have met, somewhere in their history, but that doesn't prove that they're ethnically akin at all.
After all, American and Afrikaaner settlers both used similar-looking covered wagons in their respective treks, yet one isn't necessarily descended from the other.
As for language -- according to what I've read on the net, there are some Roma loan-words in Shelta (the Traveller language), but most of the roots are Gaelic. There are also Greek and Hebrew borrowings which suggest that some of the founders of the Travellers must have been hightly educated people - perhaps the last survivors of the ancient bards of Ireland. Go back to post 40, scroll way down past the first round of references, and read the discussion of Shelta.
I heard elsewhere, that the Travellers supposedly have a higher percentage of red hair, than Ireland at large. That would strongly suggest a native insular-Celtic origin followed by genetic drift in a small, mostly closed breeding pool.
Of course, I could be wrong.