Short answer is yes.
Here are some of the the reasons.
- Flu virus vaccine is not 100% effective. It usually is about 60% effective.
- Immune system usually takes about two weeks to start defending. Exposure to virus during this phase can make you sick.
- Vaccine tends to work less well in elderly due to less efficient immune system.
- One could exposed to a strain that is not covered by the vaccine.
- A person can get exposed to other viruses that can cause flu like symptoms e.g. Adenovirus, Parainfluenza, Respiratory Syncytial Virus.