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).