Yes you can. I'm doing it all the time in my projects.
There's a storyzone object in FPSC, which allows you to attach a movie or picture to it, to tell your story. Unfortunally, at the moment there's a bug in it, but TGC is working on an update.
As for characters talking and such, I've created a script for that which is in the scripting section, where you can also find a workaround the the storyzone bug. This workaround only works with pics at the moment, not with movies, I have't looked into a workaround for that, since I don't use movies myself for my projects.
You cannot have ingame cutscenes in FPSC as some games do have. You can only use externally created pics and movies.
I'm working on a game right now which tells a story, besides having mission messages and obtaining vital info from characters also.