I know a lot of celebrants would prefer to think that. Christmas traditions as celebrated in the west are largely based on numerous pagan holidays and practices, including, but not limited to, Saturnalia. People who think otherwise simply live in the fantasy land of "coincidence theory" because of their culture - it's easier to.
I read an interesting article recently that suggested the pagan festivals were actually copies of an early guesstimate of when Christ was born, although the guesstimate was based on a tradition along the lines of prophets dying on the same day as their birth or their conception. The gist of it was that the date of Jesus' death was pegged around March 25, tradition held that he was conceived on the same day, add nine months for gestation and you end up with December 25 as his date of birth.
I don't recall the article indicating what, if any, basis there was for the tradition holding that death and conception were the same day but it cast doubts on whether or not there was a pagan festival held on December 25 prior to it being selected as the date for Christ's birth.
In many ways the whole idea is academic, since Jesus never told us we were supposed to celebrate his birth and the exact date doesn't seem to be recorded. Then there's the issue of whether the best way to "keep Christ in Christmas" is to fight endlessly over the precise name of it, or to get busy doing the things Jesus told us to do (loving each other, looking after the needy etc).