Let $C_1,\dots C_m$ be a family of ordered set partitions of $[n]$ with exactly $k$ blocks.

Write $C_i = \{B_{i1}, \dots, B_{ik}\}$ for $i=,1\dots, m$ where $B_{ij}$ are the blocks of the ordered set partition $C_i$.

Suppose this family also has the property that for each $j=1,\dots, k$

$$B_{1j} \cup \cdots \cup B_{mj}$$

is also a partition of $[n]$

Can one determine the maximal number of members in such a family $m$, or at least a decent upper bound on $m$?

Edit:

It might also be worth noting that if we take $k=n$, then $m=n$ since this would be equivalent to the existence of a latin square. I am in particular interested in the case $k=2$.