Radio-frequency-induced evaporative cooling of Rb-87 atoms in the F = 2, m
= 2 ground state in a magnetic trap with a high bias field is found to be h
ampered, and eventually interrupted, by the presence of nonlinear Zeeman sh
ifts. An experiment studying evaporation at a fixed adjustable rf frequency
was used to illustrate this effect. These nonlinear shifts prevent us from
reaching quantum degeneracy in the F = 2, m = 2 ground state; however, eva
poration in the F = 1, m = -1 ground state is not adversely affected, and w
e can produce a Bose-Einstein condensate (BEC). Our analysis shows that thi
s effect can play a role in many BEC experiments. [S1050-2947(99)50909-8].