Incidentally, a good example of a large population that gets both Medicare and Medicaid is the large population of persons living in nursing homes who cannot afford to pay the nursing home out of their own money.

Medicare will pay for stays in a nursing home, only if it's for some acute medical condition, like recovering from big-deal surgery. If you need nursing home placement because you are just unable to do ordinary things like bathe and feed yourself, perhaps - say - due to dementia, then you must pay for that. (or, someone, like a rich relative, must pay for you.) If you absolutely can't pay for a nursing home, yourself, and it would be unsafe for you to be left alone at home, then you qualify for your state's Medicaid - full benefits. If you already had Medicare, you would keep that and, also, have full Medicaid. Medicaid would pay for the nursing home care. It would also supplement Medicare in paying for your doctor and prescription bills.

Needless to say, being in a nursing home on Medicaid can, possibly, be an unfortunate way to end up in life. The quality of that care can vary hugely from state to state.