It sounds like you're talking about a schedule change. Generally, if the schedule change is not acceptable to you, you can get a full refund with no penalty. In the case of an award ticket, it would be a mileage redeposit and your taxes back.

Of course, there needs to be some sort of reasonableness applied. If they change your flight by 15 minutes, it'll be difficult to argue the schedule change was not acceptable to you. But if a flight is cancelled and your new schedule involves now a connection instead of non-stop, and 4 additional hours of travel time, then that should be a slam dunk.