One of my research problem can be reduced to a question of the following form

Given a set family $\mathcal{F}$ of $[n]$ , such that every element of $[n]$ lies in exactly $K$ sets in $\mathcal{F}$, can w partition $\mathcal{F}$ into $K$ subfamilies $\mathcal{F}_i$ such that each subfamily is a partition of $[n]$?

I feel this kind of question is very natural, and greatly appreciate any reference on this type of question.

Edit: It looks like the problem as stated admits an easy counterexample. I have posted a more interesting version here.