After years of research, I haven't been able to come up with a final solution:

According to the Oxford English Dictionary: fed up with

According to wordreference.com: fed up with BUT: in the forums there are native speakers who state that fed up of is correct in colloquial English, while others say it is not, and give the example of sick of.

According to english-test.net: fed up with http://www.english-test.net/forum/ftopic21244.html

There are, however, prestigious websites which include the expression "fed up of": see the heading of this BBC article: http://news.bbc.co.uk/sport2/hi/diving/8637305.stm

So, it seems that fed up with is absolutely correct, and fed up of might be correct. My advice is clear: fed up with!!!!!!!!!!!!!!!!!!!!!!