The classical James constant and the \(n\)th James constants, which are measure of \(B\)-convexity for the Cesàro sequence spaces ces\(_{\text{p}}\) and the Cesàro--Orlicz sequence spaces \(\text{ces}_{\text{M}}\), are calculated. These investigations show that \(\text{ces}_{\text{p}}\), \(\text{ces}_M\) are not uniformly non-square and even not \(B\)-convex. Therefore, the classical Cesàro sequence spaces ces\(_{\text{p}}\)p are natural examples of reflexive spaces which are not \(B\)-convex. Moreover, the James constant for the two-dimensional Cesàro space \(\text{ces}_2^{(2)}\) is calculated.