  The Bollob\'as set pairs inequality is a fundamental result in extremal set
theory with many applications. In this paper, for $n \geq k \geq t \geq 2$, we
consider a collection of $k$ families $\mathcal{A}_i: 1 \leq i \leq k$ where
$\mathcal{A}_i = \{ A_{i,j} \subset [n] : j \in [n] \}$ so that $A_{1, i_1}
\cap \cdots \cap A_{k,i_k} \neq \emptyset $ if and only if there are at least
$t$ distinct indices $i_1,i_2,\dots,i_k$. Via a natural connection to a
hypergraph covering problem, we give bounds on the maximum size
$\beta_{k,t}(n)$ of the families with ground set $[n]$.
