The constrained Cramér-Rao bound (CCRB) is a mean-squared-error (MSE) lower bound for non-Bayesian constrained parameter estimation under some unbiasedness conditions. In this paper, we demonstrate limitations of this bound in the case of nonlinear parametric constraints. We consider the problem of constant modulus signal estimation. It is shown that in this problem the CCRB unbiasedness conditions are too restrictive and that the commonly-used constrained maximum likelihood (CML) estimator does not satisfy them and has lower MSE than the CCRB. An alternative lower bound, which is based on the Lehmann-unbiasedness conditions, is used as an alternative benchmark for constrained parameter estimation. As opposed to the CCRB, it is shown that this alternative bound is valid for the CML estimator in the considered problem.