There are going to be a few cutscenes in Resident Evil 6 - 4 hours and 15 minutes of them, to be precise. This information comes courtesy of the British Board of Film Classification (BBFC), which serves as the UK equivalent of the ESRB in the United States. Noting that Capcom's latest action/horror game features "strong violence and gore," the BBFC gave RE 6 an '18' (think 'M').
For comparison, Resident Evil 5 and Resident Evil 4 both feature approximately one hour of cinematics each, but there is an important distinction to be made here. Resident Evil 6 contains three separate main campaigns, with a fourth that somehow unlocks later on.
Capcom has publicly stated that the three main storylines are separately not quite as long as RE 5, but collectively they are considerably longer. Taking that into consideration, and dividing the content over the span of three or four cutscenes, and what might at first seem like an interactive movie might be a little less so.