  A Griffiths phase has recently been observed by Monte Carlo simulations in
the 2D $q$-state Potts model with strongly correlated quenched random
couplings. In particular, the magnetic susceptibility was shown to diverge
algebraically with the lattice size in a broad range of temperatures. However,
only relatively small lattice sizes could be considered so one can wonder
whether this Griffiths phase will not shrink and collapse into a single point,
the critical point, as the lattice size is increased to much larger values. In
this paper, the 2D eight-state Potts model is numerically studied for four
different disorder correlations. It is shown that the Griffiths phase cannot be
explained as a simple spreading of local transition temperatures caused by
disorder fluctuations. As a consequence, the vanishing of the latter in the
thermodynamic limit does not necessarily imply the collapse of the Griffiths
phase into a single point. In contrast, the width of the Griffiths phase is
controlled by the disorder strength. However, for disorder correlations
decaying slower than $1/r$, no cross-over to a more usual critical behavior
could be observed as this strength is tuned to weaker values.
