I like the basic idea and it has been used in movies (Run Lola Run) or even games (resident evil, kind of in SC I).
The trick is,that even when using the same timeline you should deliver only a part of the story per playthrough with overlapping events, which will start as random event the player will not really recognize as story plot at first , but will resolve as clever story plot later. But avoid redundant story playthrough, the story telling should show really unique views of the same story which construct to an unique experience once you see the whole picture (after playing all game sessions).
Thanks, Ashaman, that helps. I think I need to figure out a good way to balance just how much you can resolve as one character. That is, should playing through all three characters be considered a requirement to reaching some sort of satisfying conclusion or should the player be able to finish one playthrough with Character A, B, or C and be satisfied with the ending, then play the others if they want to gain more interesting but not necessarily required insights into the overall events of the timeline? This is all in terms of revealing the overall story of this journey - each character will of course have their own missions that are relevant to their individual narrative and progression through this timeline.