As far as I know, no reliable evidence has been found to place the Coolidges in Massachusetts prior to 1636.
On 25 May 1636, John Coolidge was admitted a freeman of Massachusetts Bay Colony [Records of Massachusetts Bay Colony, 1:372].
He first received land in Watertown--thirty acres in the First Division of Lots--25 July 1636 [Watertown (Mass.) TR ("Lands, Grants and Possessions"), 1:3].
He signed on or before 18 Aug. 1636 the "Covenant of Dedham," and was a proprietor, though not a resident, of that town as late as 28 Aug. 1638, when "John Hunting enterteyned to purchase John Coolidge his Lott & crtifficate pduced" [Dedham (Mass.) TR, 3:48].
The first of his children to be recorded in Watertown, Mary, was born 14 Oct. 1637 (her brother Nathaniel, who deposed in 1678 at age forty-two years, and thus born about 1635-36, was not recorded) [Watertown (Mass.) VR, 1:5].
It seems likely that the Coolidges arrived in New England in 1635 or early in 1636.