I've never had a problem even after the cancellation period has expired.
Once, on an award stay (1 free night & 1 cash + pts), my plans changed and I wanted to change the dates a few days before I arrived. I just called the main reservations line (not the hotel itself). I mentioned that it was after the cancellation deadline and they just said, don't worry, we'll waive it. This was for a hotel with a 7 day cancellation policy.
Once I just checked out of a 7-day stay after 2 nights and went to a different hotel because I didn't like the hotel. I just asked at the front desk if there would be a fee and they said no (though, that was not an award stay)
Overall, I've had pretty good luck with Hyatt customer service. I just politely ask for what I want and they give it to me. Obviously I'm sure there's a limit (I don't think that any of my requests were particilarly unreasonable) but it's more than I'm entitled to by the rules.