You have the theorem wrong and that gives your answer. The *lower* bound is $n$ and can always be achieved. The upper bound occurs for a complete graph with each line having $2$ points. Now it is an interesting question if each line needs at least $3$ points. I believe that then for $n$ lines one must have a projective plane with $n=k^2+k+1. $ This is possible for $k$ a prime power and not known to occur on any other cases.