Shortly after my dates were released I wanted to book a flight with an award ticket for March. There were no rewards available and the price was almost $300 from Long Island to Orlando (per person, one way). I kept looking for alternate dates, but could not find anything that would work for me. I finally did manage to snag the return trip using an award.
After several weeks I was getting nervous and finally the flight down dropped to $239. No reward seats were available, so I booked that flight with cash. I kept checking the price and it came down again so I re-booked. Early this week the flight was available with a reward ticket.
In short, the answer is yes, they sometimes do become available close to the travel date. But I would not count on it happening because you never know.