I played on my 600+ cp char and got only 1 or 2 boxes after each bg game, if my team was on 3rd place I had only 1 box. But today I created new char and after each bg game, even if my team on 3rd place, I always had 3 reward boxes
Level 50 characters get a reward box every 20K AP, while level 10 get a box every 5K. As the character level goes up the AP needed to get a reward goes up.