"JigSaw Puzzle" on Set Family

Question

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 $[n]$?

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

Answer 1

No. E.g., let $n=3$ and $\mathcal F=\{\{1,2\},\{1,3\},\{2,3\}\}$, so that $K=2$. However, no subset of $\mathcal F$ is a partition of $[n]$.