With our current knowledge, we don't have a way to absolutely prove the age of the universe, so yes it is possible that either one of the estimates could be correct. However, I would say that it takes way more faith to believe in 6000 years than billions of years, because the billions of years estimate is based on the current state of our knowledge.
To put things into perspective, the 6000 years estimate comes from a time when we still thought that the earth was flat. We've made quite a few advances since then. If you're interested, this Wikipedia entry about the
age of the universe does a good job of explaining our current best estimate of the age of the universe from an empircal point of view.