The Coreys really are a Mayflower family:http://www.mayflowerfamilies.com/Salem%20Witches/martha_corey.htm

 

Bromfield Corey is said to derive from the lineage made famous by Giles Corey, who was pressed to death in Salem during the witch trials there for being stubborn. The reference to not being able to put a nail into the timbers of his ancestral home is perhaps a nod to this and a more general acknowledgement of the historical, well-known and respected nature of his breeding. This mention of his heritage also helps, in a subtle way, to insinuate the differences between his storied bloodlines and the obscurity from which the Laphams have recently risen. Also, after the pointed cuts at the style of his house, Bromfield probably felt the need to put his guests in their places once more and reiterate the refined background from whence he had come.