A cornerstone of modern polymer physics is the `Flory ideality hypothesis which states that a chain in a polymer melt adopts `ideal random-walk-like conformations. Here we revisit theoretically and numerically this pivotal assumption and demonstrate that there are noticeable deviations from ideality. The deviations come from the interplay of chain connectivity and the incompressibility of the melt, leading to an effective repulsion between chain segments of all sizes $s$. The amplitude of this repulsion increases with decreasing $s$ where chain segments become more and more swollen. We illustrate this swelling by an analysis of the form factor $F(q)$, i.e. the scattered intensity at wavevector $q$ resulting from intramolecular interferences of a chain. A `Kratky plot of $q^2F(q)$ {em vs.} $q$ does not exhibit the plateau for intermediate wavevectors characteristic of ideal chains. One rather finds a conspicuous depression of the plateau, $delta(F^{-1}(q)) = |q|^3/32rho$, which increases with $q$ and only depends on the monomer density $rho$.
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