  We consider separable 2D discrete Schr\"odinger operators generated by 1D
almost Mathieu operators. For fixed Diophantine frequencies we prove that for
sufficiently small couplings the spectrum must be an interval. This complements
a result by J. Bourgain establishing that for fixed couplings the spectrum has
gaps for some (positive measure) Diophantine frequencies. Our result
generalizes to separable multidimensional discrete Schr\"odinger operators
generated by 1D quasiperiodic operators whose potential is analytic and whose
frequency is Diophantine. The proof is based on the study of the thickness of
the spectrum of the almost Mathieu operator, and utilizes the Newhouse Gap
Lemma on sums of Cantor sets.
