Summary: We study the \(n\)-existential closure property of the block intersection graphs of infinite \(t\)-\((v,k,\lambda)\) designs for which the block size \(k\) and the index \(\lambda\) are both finite. We show that such block intersection graphs are 2-e.c. when \(2 \leq t \leq k-1\). When \(\lambda = 1\) and \(2 \leq t \leq k\), then a necessary and sufficient condition on \(n\) for the block intersection graph to be \(n\)-e.c. is that \(n\leq\min\{t, \lfloor(k-1)/(t-1)\rfloor+1\}\). If \(\lambda \geq 2\) then we show that the block intersection graph is not \(n\)-e.c. for any \(n \geq \min\{t+1,\lceil k/t\rceil + 1\}\), and that for \(3 \leq n \leq \min\{t, \lceil k/t\rceil\}\) the block intersection graph is potentially but not necessarily \(n\)-e.c. The cases \(t = 1\) and \(t = k\) are also discussed.