Yes, you can be infected with multiple viruses at once, this is called co-infection. It is currently unknown how common co-infection is with COVID-19. COVID-19 virus, the virus that causes the flu, and viruses that cause the common cold are all different viruses and there are clinical tests that can differentiate between them.

References

https://wwwnc.cdc.gov/eid/article/26/6/20-0299_article https://www.medrxiv.org/content/10.1101/2020.02.29.20027698v2https://www.medrxiv.org/content/10.1101/2020.02.12.20022327v2.full.pdf+html

Drafted 20 March 2020