Infections with antibiotic-resistant bacteria (ARB) in hospitalized patients are becoming increasingly frequent despite extensive infection-control efforts. Infections with ARB are most common in the intensive care units of tertiary-care hospitals, but the underlying cause of the increases may be a steady increase in the number of asymptomatic carriers entering hospitals. Carriers may shed ARB for years but remain undetected, transmitting ARB to others as they move among hospitals, long-term care facilities, and the community. We apply structured population models to explore the dynamics of ARB, addressing the following questions: (i) What is the relationship between the proportion of carriers admitted to a hospital, transmission, and the risk of infection with ARB? (ii) How do frequently hospitalized patients contribute to epidemics of ARB? (iii) How do transmission in the community, long-term care facilities, and hospitals interact to determine the proportion of the population that is carrying ARB? We offer an explanation for why ARB epidemics have fast and slow phases and why resistance may continue to increase despite infection-control efforts. To successfully manage ARB at tertiary-care hospitals, regional coordination of infection control may be necessary, including tracking asymptomatic carriers through health-care systems.