  We consider Schr\"odinger operators in $\ell^2(\mathbb{Z})$ whose potentials
are given by independent (not necessarily identically distributed) random
variables. We ask whether it is true that almost surely its spectrum contains
an interval. We provide an affirmative answer in the case of random potentials
given by a sum of a perturbatively small quasi-periodic potential with analytic
sampling function and Diophantine frequency vector and a term of Anderson type,
given by independent identically distributed random variables. The proof
proceeds by extending a result about the presence of ground states for atypical
realizations of the classical Anderson model, which we prove here as well and
which appears to be new.
