I was just going through the Tower of Darkness and I was like, "Shoot, I can't access my boxes, I have nothing to carry Kubfu." That got me wondering where else can't you access your Boxes at. Thanks in advance!
Gyms battles, Gym Challenges, The two towers during the challenge, anywhere you can't access your X menu(like a cutscene). Those are pretty much the only places where you can't access your boxes.