The measles vaccine is among the more effective ones, but even it isn't 100% effective in preventing symptoms. In a major outbreak of any infectious disease there will always be some vaccine breakthrough cases.
We're kind of getting into the semantics of how "infection" is defined. From a medical standpoint, an infection usually means that there is ongoing reproduction of the pathogen within a multicellular host organism. If the virus is inside some cells but not actively reproducing at detectible levels, then that wouldn't technically be classified as an infection.
Definition of infection in the Medical Dictionary by The Free Dictionary
medical-dictionary.thefreedictionary.com
Biologists working at the level of individual cells might have a different definition, I'm not sure.