Immigrants seeking health care, especially those without some kind of public or private insurance, highlight the barriers to access that arose as intended or unintended barriers of how dominant stakeholders shaped American medicine. This paper draws on a new study of those consequences for immigrants and focuses on efforts by one state to increase access. Such efforts are framed and constrained by past institutional developments and the layered actions of federal, state, and sometimes county or city actions. We develop a conceptual framework based on Merton & Barber, Meyer & Zucker, Tilly, and Massey that is useful for analyzing health care and other human service programs. Categorical inequalities underlie institutional ambivalence in many programs and policies, and in efforts to reduce or increase them. These inequalities and ambivalence contribute to American health care and health insurance being permanently failing systems driven by provider and insurer moral hazard that never collapse but run inefficiently, ineffectively, and inequitably.