Sometimes our group has had the same problem...no available campsite because all were taken. I have only had this problem in the BWCA at very busy areas in mid-summer and August which seems to be the busiest season.
We have paddled hours and miles past where we had planned to stop, just looking for an available site. We recently had to do this twice, going in and coming out, on the west end of Knife Lake. There we "raced" a pair of Forest Service people to the second to last campsite, but lost

and finally ended up at the very last campsite of over a dozen possible.
Other places we've had the same problem were Basswood River, top and bottom areas, and LacLaCroix near the Nina Moose route in/out. On this route Agnes Lake is also often filled up.
The best solution is to start out earlier in the day so you can stop earlier in the afternoon. I see a lot of people settling on a new site as early as 3:00 in the afternoon.
I keep an eye out for the numbers of canoe parties both ahead of me and behind me during the day. If it is looking very busy around me, I start looking for a site sooner rather than later during the day. When passing, I often will ask someone coming from the area of our destination if they have seen any open campsites on the way.