It’s not common but reinfection with influenza can definitely occur, either due to the same viral strain, or due to a different one altogether.

One study reported influenza reinfection due to H1N1 in otherwise healthy patients within 12-20 days of the original infection after an apparent period of full recovery. ¹ There was no evidence of resistance to oseltamivir among isolates and all patients recovered after the second infection.

Reinfection with the same viral strain within 2-3 weeks of the initial bout of influenza shouldn’t be too surprising since it takes 4-7 weeks for antibody response to the infection to peak. ² Reexposure to the same circulating strain of influenza virus (the season can last 6 weeks or longer) can then result in reinfection when the body hasn’t had enough time to make significant amount of protective antibodies following the first infection.

Another explanation is that more than 1 strains of influenza virus often circulate during any given season.   This places patients at risk of infection due to strains of influenza virus that do not confer significant cross-immunity between each other,  resulting in getting “the flu twice in 1 season.” ³

References

1. Perz CM, Ferres M, Labarca JA. Pandemic (H1N1) 2009 reinfection, Chile. Emerg Infect Dis 2010;16:156-57. https://wwwnc.cdc.gov/eid/article/16/1/pdfs/09-1420.pdf
2. Treanor JJ. Influenza viruses, including avian influenza and swine influenza. In: Mandell GL, Bennett JE, Dolin R, eds. Principles and practice of infectious diseases. 7^th ed. New York: Elsevier; 2010. p 2265-2293.
3. Rettner R. Can you get the flu twice in 1 season? Scientific American, LiveScience, February 4, 2018. https://www.scientificamerican.com/article/can-you-get-the-flu-twice-in-1-season/ . Accessed February 5, 2018.