Did Latin change any over the course of Rome's thousand year history? Did all social classes speak the same Latin?

Latin's rules are very static and formal, which makes sense for a language preserved in liturgical contexts and in science writing up through the 18th century. I just wonder if a modern Latin speaking time traveler, who learned it in High School, would be inteligible to all Romans. Rome existed for a very long stretch of time. We can't understand English from 1000 years ago.