The general rule is that a patient can only get one virus at a time. So the rule for 80+% of people is no, you either get the influenza virus or the SARS-CoV-2 virus, but not both. But in medicine, like in life, there are always exceptions to the general rule.
On March 11th doctors in China published a case in Emerging Infectious Disease of a patient who had an infection with both SARS-CoV-2 and influenza A at the same time.
Emerging Infectious Disease is a non-peer reviewed publication site run by the US CDC that infectious disease doctors can post cases to quickly, 24/7, to help their colleagues around the world. It is the modern equivalent of the conversations we used to have upon arriving at the hospital in the morning, over coffee, with the doctors and nurses that had just worked the night shift. Except now, with the internet, the medical coffee klatch includes every doctor in the world!
The timeline of this case is a good way to look at it:
Day 1: A 69 year old man is seen in a Beijing clinic for fever and a dry cough that began that day. He had returned the day before from a one month stay in Wuhan, where SARS-CoV-2 began. His x-ray was abnormal. Because of travel history he was isolated with the presumed diagnosis of COVID-19. Over eight hours he had three negative SARS nasal swab tests and one positive influenza A test. His diagnosis was changed to influenza A and he was sent home to rest, as is standard for the flu.
Day 7: He returned to the hospital with fever and shortness of breath. He required oxygen to help him breath. He had two more negative SARS swab tests. He worsened and was admitted with a diagnosis of severe influenza A pneumonia and put on a breathing machine.
Day 10: He improved. A flexible scope was used to go into his lungs through his mouth and get a big sample for testing. For the first time, he was positive for SARS-CoV-2 in that specimen. A sample of sputum was also positive but, at the same time, another nasal swab remained negative. This was his sixth negative nasal swab test!
Day 11: He was well enough to be transferred to a non-critical care hospital where he stayed until he recovered and went home.
Reference: Wu X, Cai Y, Huang X, Yu X, Zhao L, Wang F, et al. Co-infection with SARS-CoV-2 and influenza A virus in patient with pneumonia, China. Emerg Infect Dis. 2020 Jun [3/24/2020]]. https://doi.org/10.3201/eid2606.200299