  We perform a multimode treatment of spin squeezing induced by interactions in
atomic condensates, and we show that, at finite temperature, the maximum spin
squeezing has a finite limit when the atom number $N\to \infty$ at fixed
density and interaction strength. To calculate the limit of the squeezing
parameter for a spatially homogeneous system we perform a double expansion with
two small parameters: 1/N in the thermodynamic limit and the non-condensed
fraction $<N_{\rm nc}>/N$ in the Bogoliubov limit. To test our analytical
results beyond the Bogoliubov approximation, and to perform numerical
experiments, we use improved classical field simulations with a carefully
chosen cut-off, such that the classical field model gives for the ideal Bose
gas the correct non-condensed fraction in the Bose-condensed regime.
