How to find a 2-wise independent hash family that is not 3-wise independent?
The smallest example I could find is
$$
0001 \\
0010 \\
0100 \\
1000 \\
0011 \\
0101 \\
0110 \\
1001 \\
1010 \\
1100 \\
1111 \\
1111
$$
I'll leave you to generalize this to larger $n$.