the nursing home did have to let my mom stay there after her Medicare days ran out.

More info: 

A person who enters a nursing home on Medicare, has the same legal rights as any other resident. That means that, even if the person expected to be staying for only a short time while she received therapy, she has a right to stay as long as her care is being paid for and she needs nursing home care. If she has no funds and qualifies for Medicaid, the nursing home must let her stay and accept Medicaid payment for her care, if the home is entirely in the Medicaid program.

If some part of the nursing home is licensed for "skilled" or "intermediate" care and is not in the Medicaid program, the nursing home is called "distinct part" certified. A "distinct part" certified home must warn a resident and her family in writing, no later than when she moves in, that the home may choose not to accept Medicaid payment for the resident when her Medicare stops. If the resident is not in a part of the home that is certified for Medicaid when her Medicare stops, and the nursing home has given this written notice, the home may refuse to accept Medicaid for her, and discharge her if she cannot pay the private pay rate. If she is in a "Medicaid bed," or they did not give the written notice, the home may not make her leave.