When we've used our miles for Europe, generally they were for upgrades to next class of service. However, once I used them for a free ticket and booked 3 months in advance. I travelled in June, which, while a busy travel time, is not as busy as in July and August.

But if you want four tickets, then you may be asking for about 50% or more of the free seats available on any given flight, so in that case, you should book as soon as you can and hope your dates don't change. You may also want to give the airline the choice, if they can't find 4 tickets on the same flight, of getting 2 tickets on one flight and 2 on the next flight. I feel sure by booking early and giving American a lot of flexibility, you'll get your tickets.