I also thought at one time that our Joseph Bartlett was the son of Robert Bartlett and Mary Warren.Since Mary's father was Richard of Mayflower fame, this was pretty interesting.However, my sister checked many of the records documenting Mayflower descendants and disproved this connection.
In fact our Joseph was not an upstanding citizen.From "The Descendants of Joseph Bartlett of Newton, MA" by Aldis Hibner, 1934, p. 3-11. Born about 1630. Died 26 Dec 1702. Married Mary Waite RIN 204 on 27 Oct 1668, found in both Cambridge and Newton, MA records. Lived in part of Cambridge that later was known as Newton. In 1854 location of house in Newton was described as on the hill occupied by the Baptist Theological Seminary and about 90 rods southeast of the new railroad center at Newton Center. Middlesex County Court record on 25 Dec 1671 stated his behavior toward his wife was disgraceful as he was away from home day and night. Cambridge Town Records for 3 Mar 1672/73 noted he was in poor condition, had no house, food or clothing. On 12 Mar 1672/83 the Cambridge Second Church made a contribution toward the poor condition of his family. Town records for 12 May 1673 note contribution toward their relief. Many direct quotes from Middlesex County Court records of 1673 case between Bartlett as plaintiff and John Clarke of Muddy River (Brookline) concerning Bartlett's house being torn down for the lumber.
The Joseph who was a son of Robert Bartlett and Mary Warren was apparently a contemporary of our Joseph.
Please let me know if you have documentation that contradicts this parentage.