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Yes. For every prime power $q\ge 2$, a finite projective plane of order $q$ haspoints and the same number of lines; every two lines meet in exactly one point, and the intersection of all lines is empty. Its lines therefore form one required family on $n$ points. The near-pencil from part (ii) is another. They are non-isomorphic because every projective-plane line has size $q+1$, whereas the near-pencil has a member of size $n-1$. There are infinitely many prime powers, so infinitely many such $n$.