I started messing around with some genealogical searches after stumbling across the fact that I'm very distantly related to Charles Best (his grandfater, John Burbidge Best, was a cousin of one of my ancestors). I found out a few interesting things.
Let's start with the Burbidges first, shall we? A quick overview of the Nova Scotia male line is at a site dedicated to the Nova Scotia Eatons, maintained by a third cousin.
First of all, let me say that there is no evidence at all linking the Nova Scotia Burbidges with the Globe Burbages/Burbidges. It's the same name - James spelled it both ways in surviving documents - and the familie may have come from the same areas as the ones identified by the English census of 1881 as the main locus of the name - except that we came from the Isle of Wight (with some oral traditions of being connected more inland) and the very late traditions would put the Burbages in Warwickshire near Shakespeare (it is one of the two areas with the higest concentrations of names). But the records for our branch peter out in the early 18th century, and there's no obvious connection to bridge the intervening hundred years and the gap in space with London, where their family settled. I'm sure we have common ancestors, but it's an Anglo-Saxon surname and there's been a long time for it to develop and be spread around in that area.
2) There are more interesting connections. Let's start with the next in order: I had been told that we had Mayflower blood via settler ancestors (i.e. Americans who came up to Nova Scotia before the Revolution), so I decided to track that down.
We have several settler lines. The Clarkes/Clarks are one, but they resolve into relatively uninteresting late connections in New England. More interesting are the Rands, who came up from New England in about 1760, and who go back, via Martha's Vineyard, Connecticut, and Massachussetts, to Plymouth in the 1620s - but not quite to the Mayflower. The earliest in the New World of this line is a James Doughty/Doty (who shares a last name with a Mayflower passenger, Edward Doughty/Doty - actually an indentured servant who had problems with the law and fought the first duel in New England), who was born in Plymouth in 1624, was a soldier in King Philip's War, and settled back in Plymouth. However, he's either unrelated to the Mayflower man or (my guess, if the location of birth is correct) is a filius nullius who took his father's name.
There are some other prominentish names in that cluster, by early New England standards, but none which have any chance of Mayflower connections.
3) I'm going to return to the settler link on that line, though. The man who actually came up from New England in the 1760s married a woman, also from New England, who was descended from Doughty; his name was Jonathan Rand. His father was a Caleb Rand who seems to have been peripatetic, and whose parents were a John Rand and Mehitable Rand, nee Call.
Mehitable is the interesting, or at least, documented, link. Her father came over from England at about the age of one year, and came from what seems to have been a prominent gentry family. There's a history of the Calls from England by a family member which records a five-generation link back from John Call (her father) to one Richard Calle, who married the daughter of one Sir John Paston. In fact, the marriage is discussed - not positively - in the Paston letters: he was Paston's bailiff and the family was in trade (grocers, though not minor: a neice married one Henry Bacon, also a grocer, who was twice Mayor of Norwich. (There is a much more extensive history which this short document seems to have drawn on, but which I haven't looked at in detail -- this one has no interest in any American branch of the family.)
The Calls may also have a separate branch, a first cousins of the one who came over on my line, who was a Mayflower pilgrim, Thomas Rogers -- his parents (Thomas Matthew Rogers and Alice Calle) were contemporaneous with Shakespeare in Stratford, but as they were Puritans and John was a recusant, I have to assume that they were not on good terms. However, there may have been two Alice Calles with close but not identical dates -- there is little detail in the Call text regarding Alice -- in which case the identity does not hold.
The Pastons, to get back to them, were genuinely prominent and well-documented. They were Norman-descended gentry with some noble ancestors, which include the de Glanvilles - yes, including the justiciar and legal author - and via them there are links to Norman nobility - they really did come over with the Conquereror, at which time the Pastons are already prominent and recorded in the Domesday Book - and via them to both Danish Jarls a-Viking, Dukes of Normandy, and Kings of Brittany and Italy, and so, ultimately, to Charlemagne.