"being full to the brim," 1660s, present participle adjective from brim (v.).
c.1200, brymme "edge of the sea," of obscure origin, perhaps akin to Old Norse barmr "rim, brim," probably related to German bräme "margin, border, fringe," from PIE *bhrem- "point, spike, edge." (Old English had brim in the sense "sea, surf," but this probably was from the Germanic stem *brem- "to roar, rage.") Extended by 1520s to cups, basins, hats.
"to fill to the brim," 1610s, from brim (n.). Intransitive sense ("be full to the brim") attested from 1818. Related: Brimmed; brimming.