The influenza vaccine cannot cause influenza (the flu) if you're vaccinated with the inactivated trivalent vaccine, made with killed virus. However, fever and achiness can occur after a flu vaccine. This is not the flu, however, but the result of an activated immune system.
The nasal flu vaccine, which contains a weakened live virus, could, conceivably, cause the flu in someone with a suppressed immune system. Thus, it is only approved for use in healthy people between ages 2 and 49 (younger and older people tend to have weaker immune systems). Studies involving hundreds of healthy children and adults showed no evidence that the nasal flu vaccine resulted in the flu.
However, you can get the flu after being vaccinated if the viral types used to make the vaccine do not match the circulating flu viruses. These viruses change every year, which is why the vaccine changes every year and why you need an annual vaccine. Nonetheless, in any given year the flu vaccine typically protects about 60% of healthy adults under 65. The older you are, the less effective it is, likely because of a weaker immune system.
Even when the vaccine and viruses aren't well matched, the vaccine still protects a considerable number of people. Plus, if you get the flu, having had a vaccine can mean a quicker recovery with fewer complications. And don't forget it takes about two weeks after you're vaccinated before the vaccine fully engages your immune system. During those two weeks, you're still susceptible to an influenza virus, even one the vaccine should protect against.