Chambers Scots Dictionary:
to fill to the brim; to drink healths in a bumper

Chambers 20th Century Dictionary:
a cup or glass filled to the brim for drinking a toast; anything large or generous in measure

I seem to recall a (whaling? fishing?) song with a verse about the ship (or hold?) being bumper full -- - - -

also I've come across it in more than one song with the 'drinking' meaning -- canny remember exactly what anes jist the noo - --