Let $1\leq t\leq k\leq v$ be integers.
A Steiner system $S:=S(t,k,v)$ is a collection of subsets $K$ of size $k$ of a set $V$ of size $v$ such that for every subset $T\subseteq V$ of size $t$, there exists a unique $K\in S$ with $T\subseteq K$.
For $t=1, k=2$, it is obvious that an $S(1,2,v)$ exists if and only if $v$ is even; in this case, an $S(1,2,v)$ is simply a partition of the elements of $V$ into pairs.
Moreover, it can be shown that there exist $v-1$ Steiner systems $S(1,2,v)$ on $V$ that are mutually disjoint and whose union is the set of all $2$-element subsets of $V$. (Let's call systems of Steiner systems like this "orthogonal").
Does this behavior extend to larger values of $t$ and $k=t+1$?
A well-known fact is that an $S(2,3,v)$ exists if and only if $v$ is congruent $1$ or $3$ modulo $6$. In this case, the quotient of the number of all $3$-element subsets of $V$ and the size of one such Steiner system is $v-2$, so theoretically, there could be $v-2$ Steiner systems $S(2,3,v)$ whose union is all of $\left\{K\subseteq V|\;|K|=3 \right\}$.
Is this indeed the case, maybe at least for special values of $v$? And is anything known about "orthogonal" systems of Steiner systems for even larger values of $t$ and/or $k$?