Let (X; U ) be a random vector with spherically symmetric distribution about (ร; 0) where dim X =dim ร=p and dim U =dim 0=k (for p ยฟ 3). Consider estimation of ร with loss ร-d 2 . Fourdrinier and Strawderman (J. Multivariate Anal. 59 (1996) 109) considered two classes of James-Stein type estimators a (X; U )=(1-a[(U U ) =X X ])X for = 0 and 1. The case =0 is referred to as a "classical" James-Stein estimator and the case = 1 as a "robust" James-Stein type estimator since = 0 corresponds to the original James-Stein estimator in the normal case with known variance. In contrast, the case =1 corresponds to an estimator which, for a=(p-2)= (k + 2), simultaneously and uniformly dominates X for all spherically symmetric distributions. Fourdrinier and Strawderman showed that, for certain spherically symmetric distributions, the optimal (a = (p -2)=(k + 2)) James-Stein estimator for = 1 simultaneously dominates all James-Stein estimators with = 0. They term this situation a paradox since an estimated value of 2 in the shrinkage constant leads to an estimator which improves over the entire class of estimators which use the known value of 2 in the shrinkage constant. We show in this paper that this paradox is inevitable whenever the underlying distribution is a nondegenerate mixture of normal distributions and the dimension, k, of the residual vector U is su ciently large. We also calculate the "critical dimension" k0 for a variety of examples including the Student-t.