Josiah The Signer is not related to my Connecticut Bartlett line (descendants of William said to have arrived in CT 1694 https://www.wikitree.com/wiki/Bartlett-3244
). Although "family legend" claimed me/us as descended from Josiah, The Signer. As it turned out, we had Josiahs in the family all right, just none of them turned out to have been Josiah Bartlett, The Signer!
But William, as far as I know (or don't know) was not related to the Robert you mention. There were maybe 5 (or more) early CT Bartletts, not known to have been related to each other. Although some have occasionally been suggested to have been related. So some early CT Bartletts heads of family may have been related in some way to others, but probably only yDNA would be able to tell that at this point...
What I have found is that it seems like for almost every Bartlett family, someone in the family has spread a family legend that they are descended from The Signer. And it's almost never true. But that hasn't stopped quite a few of them from "creating" a relationship by creating inaccurate family trees. I know of quite a few such instances. The few to whom I've mentioned where they went astray as to parentage along the way in their tree, they don't seem to be able to consider the possibility of any error that might remove them from having been descended from The Signer. At least that seems to be the case for those with family trees in Ancestry, which is so rife with major errors from one person's mistake that spreads like wildfire onto hundreds of others' trees, being copycatted without anything being checked.
There are probably far more distinct Bartlett lines than people generally imagine. I think the last count I looked at, per yDNA, was something like 25 separate lines that could-not-have-been-related in thousands of years. And that only represents lines where someone has been yDNA tested! :-)