First, let me say that it is very common for Bartlett people to claim they descend from Josiah, the signer. Many times they are wrong. So the best way to determine any ancestry is to go from the known to the unkown.
I find Alden and Mary Bartlett in the 1920 U.S. Census, Asbury Park, New Jersey. Alden is shown as age 24 born in NJ, but his father (not named) is shown as born in Massachusetts. That is a good sign, since Josiah is from New Hampshire, next to Massachusetts.
Alden is shown with your father George age 1 year, 10 months and his brother Francis 7 months.
So your work is cut out to search for Alden's father.
I maintain a database of Richard Bartlet descendants (Josiah is one of them). I have no record of your grandfather Alden. You can access my database with sources cited at: