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We study the notion of geometrical finiteness for those Hilbert geometriesdefined by strictly convex sets with $\C^1$ boundary. In Gromov-hyperbolicspaces, geometrical finiteness is defined by a property of the group action onthe boundary of the space. We show by constructing an explicit counter-examplethat this definition has to be strenghtened in order to get equivalentcharacterizations in terms of the geometry of the quotient orbifold, similar tothose obtained by Brian Bowditch in hyperbolic geometry.

Finitude g\'eom\'etrique en g\'eom\'etrie de Hilbert