Summary: A graph is \(n\)-existentially closed \((n\)-e.c.) if for any disjoint subsets \(A\), \(B\) of vertices with \(|A \cup B| = n\), there is a vertex \(z \notin A \cup B\) adjacent to every vertex of \(A\) and no vertex of \(B\). For a block design with block set \(\mathcal{B}\), its block intersection graph is the graph whose vertex set is \(\mathcal{B}\) and two vertices (blocks) are adjacent if they have non-empty intersection. In this paper, we investigate the block intersection graphs of pairwise balanced designs, and propose a sufficient condition for such graphs to be 2-e.c. In particular, we study the \(\lambda\)-fold triple systems with \(\lambda \ge 2\) and determine for which parameters their block intersection graphs are 1- or 2-e.c. Moreover, for Steiner quadruple systems, the block intersection graphs and their analogue called \(\{1\}\)-block intersection graphs are investigated, and necessary and sufficient conditions for such graphs to be 2-e.c. are established.