A scene I could see playing out is where you step into the shoes of a refugee (this would be a 3rd-person view) who's encampment is under attack from ["evil" militia] soldiers and you have to escape within a certain time. But it won't be easy because you'd have a wife and child in tow and that slows your escape. So you're running with a small child in one arm, your holding your wife's hand with the other, and you need to avoid catching the enemies' attention as well as get to safety in time. First "time penalty" you'd receive is your escape doesn't go unnoticed and the soldiers are shooting and actively looking for you, which results in the character's wife being shot dead. Now you still have the child in your arms, crying and screaming, but now you can move a little faster. Second "penalty" would be that you get shot and are incapacitated, then (cutscene) you can only watch, helplessly, as the character's child is violently taken from his arms and carried off to a fate unknown, then the soldiers are preparing to execute you when rebel forces take out the soldiers and take you to safety, although you'd only catch glimpses as the characters slips in and out of consciousness (too predictable?). There wouldn't be any "Game Over", no "Restart" or "Mission Failed", you'd continue the game knowing that the character (you) failed to bring his wife and child to safety. Maybe there shouldn't be "penalties" and the event should be set pieces for the story, just to make it seem less like a video game (who wants an immersive experience to be broken because of "time limits" and "penalties"?)
This may seem too dark or perhaps too "edgy" (although I doubt it) but the game could end with a great triumph after a long road of redemption/vengence to keep it from being completely depressing.