  The origin of the non commutativity of the limits $t \to \infty$ and $N \to
\infty$ in the dynamics of first order transitions is investigated. In the
large-N model, i.e. $N \to \infty$ taken first, the low temperature phase is
characterized by condensation of the large wave length fluctuations rather than
by genuine phase-ordering as when $t \to \infty$ is taken first. A detailed
study of the scaling properties of the structure factor in the large-N model is
carried out for quenches above, at and below T_c. Preasymptotic scaling is
found and crossover phenomena are related to the existence of components in the
order parameter with different scaling properties. Implications for
phase-ordering in realistic systems are discussed.
