  Let $d \ge 2, h \ge 1$ be integers. Using a fragmentation technique, we
characterise $(h+1)$-tuples $(R_1, \dots, R_h, R)$ of non-empty families of
partitions of $\{1, \dots, d\}$ such that it suffices for an order-$d$ tensor
to have bounded $R_i$-rank for each $i=1,\dots,h$ for it to have bounded
$R$-rank. On the way, we prove power lower bounds on products of identity
tensors that do not have rank $1$, providing a qualitative answer to a question
of Naslund.
