I've always had really good luck getting tickets when I've needed to for the same day or the next day (normally these are due to family emergencies, funerals, etc). The airlines seem to really release a lot of their excess inventory in that time frame, from what I've seen. I don't think I'd try it on the Wednesday before Thanksgiving, though (i.e. there need to be at least a decent number of empty seats).

However, even this isn't ironclad. A few years ago I booked a flight using miles (At the saver award level) from San Jose->Lima, Peru the day before the flight, and it turned out that both legs were oversold. If I hadn't been in a hurry (the reason I was on the flight in the first place), it would have been a nice bonus...

Personally, I think this is the best use of miles, since even the bereavement fares are pretty expensive, compared to what you can get with an advance purchase...

Dan