

The Berners Family seems to be largely descended from the Norman lord Hugh de Berners, an Anglicised version of Hugo de Bernières. There are a number of locations with that name, but this is believed to be Bernières-d'Ailly near Caen.
To track the main Berners line and try and figure out where the other Berners lines fit in and to track down the various Barnes lines that are also descended from them.
There is a relatively solid line of descent from Hugh de Berners to Sir James de Berners (right, for the last part), thanks to the land]the family held from the time of the Domesday Book. This becomes complicated in the time of Sir James Berners as he was a favourite of the King and so fell victim to the Merciless Parliament. After his execution some of his children and close family appear to have also gone by the name Barnes (see, for example Thomas Barnes/Berners). His grand-daughter Margery Berners married Sir John Bourchier who became the first Baron Berners.
This change also seems to have happened with Edmund Barnes, although that line is currently separate from the above, it does appear to merge with the main one.
There is another line of Berners here that goes back to Sir Raphe de Berners. it looks like it could be merged but there are disparities in the trees.
The General Armory of England, Scotland, Ireland, and Wales, Comprising a Registry of Armorial Bearings from the Earliest to the Present Time by Bernard Burke (see also Coat of Arms Database) contains four coats of arms for Barneses based on the Berners blazon:
NB:
Made using DrawShield.
Testing the Y chromosome of Berners and relevant Barnes males would help us test the hypotheses that a branch of the Barneses are descended from the Berners and that the unlinked Berners line ties back into the main one at some point.
At the moment no Berners male appears to have tested his Y DNA and no known Barnes descendant of them has either. There may be members of the Barnes Project on FTDNA who are Berners descendants whose trees don't link back into the Berners one (as British records tend to thin out by the time you get back to the 1500s) but to prove this would require know descendants to test.
See also: Berners on Wikipedia
Notable places named after both named after Hugh de Berners, who is listed in the Domesday Book as owning the areas, and his descendants (some of whom became Barneses) :