The Barnum Family says that Nathan Barnum was born in 1718 and married to Mary Wells in 1726 (when he would have been eight years old). The Genealogical Record of the Barnum Family has him born in 1710, married to (1st) Rebecca Lockwood about 1731, and (2nd) Jemima Stevens. Another sources shows that a Nathan Barnum, of Danbury, married Hannah Bonnel or Bunnel on 10 Aug 1738 in New Milford, Connecticut. If she were the mother of Nathan's son Nathan, that would certainly explain why that son named one of his sons Nathan Bunnel Barnum. Clearly, however, further research is needed for this family group.
A third source says that Nathan Barnum was born in 1702.