  We study non-superposition effects in the Dirichlet Casimir interaction
energy for N boundaries in d spatial dimensions, quantifying its departure from
the case of an interaction where a superposition principle is valid. We first
derive some general results about those effects, and then show that they only
become negligible when the distances between surfaces are larger than the sizes
of each individual surface. We consider different examples in one, two and
three spatial dimensions.
