The Phrase Finder says that the expression means is to be tired of, bored with, annoyed with or in general to have enough of something or someone. There is an old English proverb that “enough is as good as a feast.” “Fed up” probably comes from the unpleasant feeling that comes from eating more than is good for us. The expression dates back to the 19th century when overfed aristocrats were compared to farm animals that were force fed to make them plump for market. There was a cynical section of an English newspaper article in The Middlesex Courier published in 1832 that argued a Duke could not have hanged himself, because he could not have possibly stood on a chair and tied the knot.  The writer observed about such aristocrats, ” Every thing being done for them, they never learn to do anything; they are fed up…”  Sometimes the expression is used in the extended forms “fed up to the eyeballs,” or “fed up to the back teeth.”