Consider the 9-point affine plane $\ F_3\times F_3,\ $ over the 3-element Galois field $\ F_3.\ $ Then $\ n=9,\ $ while the respective $\ m(C)=4 < n-1$.
Here $\ C\ $ is the set of affine lines, i.e. sets described by the linear equations (homogeneous and non-homogenous, just like in an elementary school).
The explicit set of 4 straight lines which include point (0\ 0) is:
$$ \{(0\ 0)\ \ (0\ 1)\ \ (0\ 2)\}$$ $$ \{(0\ 0)\ \ (1\ 0)\ \ (2\ 0)\}$$ $$ \{(0\ 0)\ \ (1\ 1)\ \ (2\ 2)\}$$ $$ \{(0\ 0)\ \ (1\ 2)\ \ (2\ 1)\}$$
Now let us consider a 6-point space which is a union of two parallel lines of the above plane, i.e. now $\ n=6.\ $ The in family $\ C\ $ consists of the intersections of the straight lines with our 6-point space (all members of this new space are two 3-point sets, and nine 2-point sets). Now $\ m(C) = 4 < n-1. $ This time I believe that this $\ n=6\ $ is minimal.