  We study the relation between the multigraded Castelnuovo-Mumford regularity
of a multihomogeneous ideal $I$ and the multidegrees of a Gr\"obner basis of
$I$ with respect to the degree reverse lexicographical monomial order in
generic coordinates. For the single graded case, forty years ago, Bayer and
Stillman unravelled all aspects of this relation, which in turn the use to
complexity estimates for the computation with Gr\"obner bases. We build on
their work to introduce a bounding region of the multidegrees of minimal
generators of multigraded Gr\"obner bases for $I$. We also use this region to
certify the presence of some minimal generators close to its boundary. Finally,
we show that, up to a certain shift, this region is related to the multigraded
Castelnuovo-Mumford regularity of $I$.
