When booking for a party of 6, I had no trouble getting what we wanted (I even lucked into one for the Anna and Elsa M&G) 30 days out. There were a few FP+ that I couldn't get for precisely the time I wanted, but I went ahead and booked them, and just returned later to change them to a more desirable time.
The 30 days is key. I tried to make some changes later, and found that by the time we were within a week or two of our visit, there was no longer much of any availability to change around FP+ for headliners. By the time we were on property and had to change some using the MM+ App, availability was extremely limited. In fact, after a WDW transportation breakdown delayed us for 90 minutes one morning, we canceled one set of FP+ to change them to another park, found no availability for "Tier 1" headliners at the other park, and then tried to reinstate the ones we'd canceled - in under 3 minutes, they were already gone. When we arrived at the park (Animal Kingdom) within 30 minutes of rope drop, there were huge placards at the first FP+ station we passed, informing us that all FP+ for the three major attractions were already gone (and this was in early May, when crowds were on the low-average side).
