I've had the good fortune to write two articles on the English origins of 16th century New England immigrants: The English Origins of William Dudley of Guilford, Connecticut, The American Genealogist 82 (2007):63-75 and The English Origins of Elizabeth Dowse, wife of Governor Samuel Allen of New Hampshire, The New England Historical & Genealogical Register 162(2008):164-73. It was really hard work.
Luckily in the two above-cited articles I was able to find the parents, grandparents, and great-grandparents of the immigrants in question. Some of the wives' ancestry was not pursued and can be further researched by myself or others in the future. Sometimes I think we should divide that phrase into English origins (we know whence they came) and English ancestry (we know the parents).