A year and a half ago I was watching LH business class availability LAX-FRA for a week (because I wanted to change a United award from AC connection to LH nonstop), and it was showing last minute availability for some days (including luckily for the day I was flying), but never ever showed last minute availability for some other days.
I've seen similar patterns with some other airlines (to other destinations) in some other programs.
So the more flexibility you have on the date you're willing to fly in each direction, the better, if you're intending to rely on last-minute awards.