In a measles outbreak, do children who have not had MMR vaccine pose a threat to vaccinated people? It is my understanding that vaccinated people can still contract measles. Am I correct?

You are correct that vaccinated people can still be infected with viruses or bacteria against which they are vaccinated. No vaccine is 100% effective. Vaccine effectiveness varies from greater than 95% (for diseases such as measles, rubella, and hepatitis B) to much lower (60% for influenza in years with a good match of circulating and vaccine viruses, and 70% for acellular pertussis vaccines in the 3-5 years after vaccination). More information is available for each vaccine and disease at www.cdc.gov/vaccines/by-disease/ and www.immunize.org/vaccines.