The Bollobás 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 $β_{k,t}(n)$ of the families with ground set $[n]$.