I think it's possible. Civilization is around 7000 years old. From a geological standpoint, that's nothing. The Earth is over 4 billion years old, while the Universe is over 15 billion years old. I think it's possible that if human beings are still around millions of years from now, humans can have used generational ships to reach other solar systems. Each star colony will essentially largely isolated form the other star colonies. Communications between star colonies will be possible, but will have ping times in the span of years. Each colony can then send out generational ships to other uncolonized stars. If this process repeats itself over millions or possibly even billions of years, it's possible for humans to colonize the galaxy.