  We consider de Branges-Rovnyak spaces of a considerably large class of
reproducing kernel Hilbert spaces and find a characterization for them to be
complete Nevanlinna-Pick spaces. This extends as well as recovers earlier
characterizations obtained for the Hardy space over the unit disc (\cite{Chu})
as well as for the Drury-Arveson space over the unit ball (\cite{Jesse}). Our
characterization takes a complete form for the particular cases of the Hardy
space over the polydisc and the Bergman space over the disc. We show that a
non-trivial de Branges-Rovnyak space, associated to a contractive multiplier,
of the Hardy space over the bidisc or the Bergman space over the unit disc is a
complete Nevanlinna-Pick space if and only if it is isometrically isomorphic to
the Hardy space over the unit disc. On the contrary, it is shown that
non-trivial de Branges-Rovnyak spaces of the Hardy space over the $n$-disc with
$n\ge 3$ are never complete Nevanlinna-Pick spaces.
