Although it is hard for people to believe, you *cannot* get the flu from a vaccination. The reason is that vaccines are made either from inactivated viruses (which cannot reproduce or infect anything, really) or from little 'pieces' of virus that your immune system thinks is the bug and makes antibodies against.
However, it takes two weeks for the flu vaccine to become effective, and if you are exposed to the flu either before you get the shot or during those two weeks after the shot, you can sure as heck come down with the flu. Flu has an 'incubation period,' which means that for several days before you're actually sick, you don't feel any symptoms even though you have actually been infected--and 'infective'. During that time, you can unwittingly infect others.
Second, as ed says, there are a LOT of different types of flu. Each year's vaccine contains antigens only from the types that health agencies think are the most 'likely' to do damage this time around. If you are exposed to a type that is not included in the recipe, you can of course get sick from that one.
from what I hear, they choose 6 or 7 of the most likely flu strains that will hit your nation and ones which can have the virus cultivated in an egg. The viruses are harmless though, they are not meant to harm you in any shape or form. All they are meant to do is trick your immune system into making the right shaped cells to lock onto them, and attack them. However, I guess if most of your immune system is geared to the wrong types, then you could get into trouble. I receive letters every year about having a flu jab but I ignore them. I'm a firm believer that the human immune system is finely tuned and should be left alone unless seen to be obviously struggling. I haven't had flu this year, normally have an episode every year but fight it off within 7 days.