Having stirred up discussion with New England Deans, I've moved up the ancestral line and found Deanes and Dennes in England.
The earliest are Dennes with an entry in the Visitation of Kent. The Visitation pedigree works nicely until one gets to John Denne, who is a third son. WikiTree gives him a son Michael, who is the father of John Deane who begins a family in Chard, Somersetshire.
This is of course possible. People do change the spelling of their names, especially in medieval times. And people did move from places like Kent, in the East of England, to Somerset, in the West, below Wales.
It's also possible that we're looking at one of those instances where one family got grafted on to another.
Can anyone find additional information that would help us decide which of the possibilities are more likely?