Even after receiving the flu vaccine you can get the flu if you are exposed to a different type of flu virus than was in the vaccine, if your immune system is not functioning properly, or if you are exposed before the vaccine was able to create the proper antibodies from the vaccination (usually 2 weeks after the vaccination in most adults and longer in children - up to a month) . The flu vaccine is made with the viral particles of the top 3 flu virus strains that are expected to be circulating in that upcoming season for which the vaccine was developed. US CDC scientists determine each year what viruses should be in the following year's vaccine based on what viruses are circulating in Asia, since these will be moving west toward the US during the flu season. There are many strains, if the scientists do not predict the correct ones, you may still become ill despite a vaccination.
You CANNOT get the flu from the flu vaccines. The viruses in the vaccines are killed (inactivated) or weakened, so you cannot get the flu from a flu vaccination. You can check out the NIH's website for more info.