1
general::
Also, fed to the teeth;fed up. Disgusted, unable or unwilling to put up with something. For example, I'm fed to the gills with these delays (the gills here is slang for "mouth"), or He was fed to the teeth with her excuses, or I'm fed up? let's leave right now. Of these colloquial expressions, fed up, alluding to being overfull from having overeaten, dates from about 1900, and the others from the first half of the 1900s. Also see UP TO ONE'S EARS.
American Heritage Idioms