(Auke found a counter-example to my original post, so I am opening this question up...)
Suppose we have $k$ sets $X_1, \dots, X_k$ which are subsets of a ground set $X$ of size $n$. We know that $|X_i| \geq t$ for all $X_i$. Then we would like to find the smallest possible set $Y$ such that $Y \cap X_i \neq \emptyset$ for all $i$.
Given fixed values for $k, n, t$, what is the largest $|Y|$ which might be necessary to do this?