How about $n=7$, so $N={3\choose 2} = 3$, with subsets $S_1 = \{ 1,2,3,4\}$ and $S_2 = \{ 1,2,5,6\} $ and $S_3 = \{ 3,4,5,6 \} $ and $S_4 = \{ 1, 3, 5, 7 \} $.

How about $n=7$, so $N={3\choose 2} = 3$, with subsets $S_1 = \{ 1,2,3,4\}$ and $S_2 = \{ 1,2,5,6\} $ and $S_3 = \{ 3,4,5,6 \} $ and $S_4 = \{ 1, 3, 5, 7 \} $.

Added later: this example can be modified to $n=8$ by taking $S_1 = \{ 1,2,3,4 \}, $ $S_2 = \{ 1,2,5,6\} $, $S_3 = \{ 1,2,7,8 \} $, $S_4 = \{ 3,4,5,6\} $, $S_5 = \{ 3,4,7,8 \} $, $S_6 = \{ 5,6,7,8 \} $ and $S_7 = \{ 1,3,5,7\} $ while $N=6$ in that case.