  We prove that certain families of homogenous affine iterated function systems
in $\mathbb{R}^d$ have the property that the open set condition and the
existence of exact overlaps both occur densely in the space of translation
parameters. These examples demonstrate that in the theorems of Falconer and
Jordan-Pollicott-Simon on the almost sure dimensions of self-affine sets and
measures, the set of exceptional translation parameters can be a dense set. The
proof combines results from the literature on self-affine tilings of
$\mathbb{R}^d$ with an adaptation of a classic argument of Erd\H{o}s on the
singularity of certain Bernoulli convolutions. Our result encompasses a
one-dimensional example due to Kenyon which arises as a special case.
