  After a general discussion about the quantitative meaning of the naturalness
upper bounds on the masses of supersymmetric particles, we compute these bounds
in models with gauge-mediated soft terms. We find interesting upper limits on
the right-handed slepton masses that, unless the messenger fields are very
light, disfavor minimal models with large messenger content. Deep unphysical
minima, that however turn out to be not dangerous, are usually present in such
models. The mu-problem can be solved by adding a light singlet only at the
price of a large amount of fine tuning that gives also rise to heavy sparticles
and large tan\beta.
