I'm not a vet but I'd say they have just as much chance of becoming ill a rabbit of any age. Flora and Parsnip were nearly 6 when I adopted them and they've be no less healthy than any of my other bunnies (past and present).
I guess that once they become ancient and their organs are under stress just living from day to day, then they become more prone to picking up illnesses or are less able to recover from them than a younger, more robust bunny.