I don't think it can be infinite.
Consider 2^n n element sets. A1, A2, ...., A(2^n). Now consider the sets A1 intersection Ai i= 1,2,3,...2^n
These are all subsets of A1. But there are only 2^n subsets of A1 (or ant n element set) So if all these interesctions are distint one of them must be the empty set. If they are not distict that is if A1 intersection A2 and A1 intersection A3 are the same then A1 intersection A2 intersection A3 is equal to A1 intersection A2.
Anyone know if (2^n) -1 is the strictest upper bound possible?