Is it normal to have false memories during a dream?

People often speak of dreaming that they're in a certain place or time, but I just as often dream of being other people entirely. Not specific people, mind you, just not myself. As part of the package, that means having their memories.

For example, if I dream that I'm in an action sequence fighting some monsters, I know exactly why I'm doing it intrinsically, what they're called, etc. If I dream I'm talking to someone, I know their name. If I dream of arriving somewhere, I can actually remember how I got there even if the trip wasn't part of the dream itself. Not always, but you get the idea.

Has anyone else experienced this sort of thing?