c.1200, brymme "edge of the sea," of obscure origin, perhaps akin to O.N. barmr "rim, brim," probably related to Ger. bräme "margin, border, fringe," from PIE *bhrem- "point, spike, edge." (O.E. brim meant "sea, surf," and probably was from the Gmc. stem brem- "to roar, rage.") Extended by 1520s to cups, basins, hats.
"being full to the brim," 1660s, from prp. of brim.