  For a scheme of fat points $Z$ defined by the saturated ideal
$\mathcal{I}_Z$, the regularity index computes the Castelnuovo-Mumford
regularity of the Cohen-Macaulay ring $R/\mathcal{I}_Z.$ For points in "
general position" we improve the bound for the regularity index computed by
Segre for $\mathbb {P}^2$ and generalised by Catalisano, Trung and Valla for
$\mathbb {P}^n$. Moreover, we prove that the generalised Segre's bound
conjectured by Fatabbi and Lorenzini holds for $n+3$ arbitrary points in
$\mathbb {P}^n$. We propose a modification of Segre's conjecture for arbitrary
points and we discuss some evidences.
