And dreams can be "lucid."
Just today, I was talking with a friend who mentioned a famous politician she had once met. Suddenly, I was flooded with a FULL MEMORY of meeting an important person in a Presidential Administration, but suddenly realized it was the memory of a dream!
There was another person I had ACTUALLY met, but the dream "important official" didn't quite have a name because, well, he never actually existed.
We don't notice the little tip-offs when we're IN a dream, but we can train ourselves to do so.