"brink, edge, margin," c. 1200, brymme "edge (of the sea), bank (of a river)," a word of obscure origin, chiefly Northern, which is probably from or related to dialectal German bräme "margin, border, fringe," from PIE *bhrem- "point, spike, edge." It was extended by 1520s to the upper or projecting edge of anything hollow (cups, basins, hats).

Old English (and northern Middle English) had brim "sea, surf, pool, spring, river, body of water," of uncertain origin perhaps akin to Old Norse barmr "rim, brim." "It became obs. in ME.; but was perhaps used by Spenser" [OED].

"to fill to the brim," 1610s, from brim (n.). The intransitive sense of "be full to the brim" is attested from 1818. To brim over "overflow" is from 1825. Related: Brimmed; brimming.

as a style of hat, 1680s, from broad (adj.) + brim (n.). Broad-brimmed in 18c.-19c. suggested "Quaker male," from their characteristic attire.

"being full to the brim," 1660s, present-participle adjective from brim (v.).

seaport city in northern Germany, from Old Saxon bremo "edge" (related to English brim (n.)), in reference to its site on a river bank.

also brim-full, "full to the top," 1520s, from brim (n.) + -ful.

"narrow ledge," 1729, from French berme (17c.), from Old Dutch baerm "edge of a dike," which is probably related to brim (q.v.). In U.S., especially "grass strip beside a road," originally the name for the bank of a canal opposite the tow path (1833; berm-bank is from 1832).

type of soft felt hat with a curled brim and a dented crown, 1894, from Homburg, resort town in Prussia, where it was first made. Introduced to England by Edward VII.

Middle English overflouen, from Old English oferfleow "to flow across, flood, inundate," also "to flow over (a brim or bank);" see over- + flow (v.). Common Germanic (Old High German ubarfliozan, German überfliessen, etc.). Related: Overflowed; overflowing.

"having the feathers of the top of the head elongated and conspicuous," 1728, from Latin pileatus "capped," from pileus "conical felt cap without a brim," which is perhaps from Greek pilos "felt; felt hat," also "felt shoe, felt blanket," or they may be from a common source (somewhat similar words are found in Germanic and Slavic). Beekes calls it "an old culture word of unknown origin." Applied in natural history to sea urchins and certain birds, notably the pileated woodpecker, a large species of North America.