Cubic spline interpolation provides a good and handy method to approximate a given function or to fit a given set of points. However, such an interpolation process does not always converge. It is known that the local mesh ratio (that of the lengths of two consecutive intervals) is less than 3 + sq. root/2, the interpolation process works for any given bounded data. This paper continues such investigation. It is shown that the above restriction on the knots may be relaxed. Thus, for a wider class of knot sequences, the cubic spline interpolation can be still applied. Hopefully, this would make such interpolation process more feasible in practice. (Author).