Nurses Certification And Recognition of Experience Act of 2021 or the Nurses CARE Act of 2021 This bill expands qualifying training and competency requirements for nurse aides in Medicare skilled nursing facilities (SNFs) and Medicaid nursing facilities (NFs) for nurse aides who initially train and work during the public health emergency relating to COVID-19 (i.e., coronavirus disease 2019). Currently, SNFs and NFs may not hire nurse aides for longer than four months unless they meet certain training and certification requirements. The Centers for Medicare & Medicaid Services has waived some of these requirements during the emergency period so as to allow facilities to maintain adequate staffing levels. The bill allows individuals who complete a training program, competency assessment, and at least 80 hours of supervised work experience as a temporary nurse aide during the emergency period, and who meet other specified competency requirements, to qualify for prolonged work as nurse aides in SNFs and NFs beyond the emergency period.