Conclusion
It is possible to be covered by both Medicare and private health insurance. Medicare is given by the federal government while private health insurance is given mostly to employees of a private companies. They help to cover the cost of medical services that you receive. However, you should know that they might not cover all of your expenses, you might need to pay a little percentage from your pocket.
Anyone of them can be your primary payer depending on some set of conditions. If private insurance fails to pay for the medical expenses when they are the primary payer, Medicare can help you pay conditionally and wait for a refund from your private insurance company.
References
[1] https://www.medicare.gov/what-medicare-covers/what-part-a-covers
[2] https://www.medicare.gov/what-medicare-covers/what-part-b-covers
[3] https://www.medicare.gov/sign-upchange-plans/types-of-medicare-health-plans/medicare-advantage-plans
[4] https://www.ncoa.org/article/what-is-medigap
[5] https://www.healthline.com/health/medicare/tricare-for-life-and-medicare-advantage
[6] https://www.mayoclinic.org/diseases-conditions/end-stage-renal-disease/symptoms-causes/syc-20354532