This paper deals with the problem of finding error bounds for cubic spline interpolation of functions of the classC 4[a, b], andC 5[a, b], by examining a relationship between cubic spline interpolation and piecewise cubic Hermitian interpolation. The method also gives an indication of what happens, in the case of almost uniform meshes, especially if the approximated function is in the classC 5[a, b]. Comparison is made with recent work carried out by K. E. Atkinson [3], in dealing with natural cubic spline interpolation.