I will go out on a limb and pretend I understand the question and suggest the following for an answer.
Mark's example of 1x2 bricks in two dimensions can be modified to use 1x1 bricks, perhaps at the expense of his constraint on the distance between disjoint blocks. For 3 dimensions, it should be clear that alternating layers of the same 2d pattern of cubes can be arranged so that every point belongs to at most 4 bricks, using bricks of length 3.
I submit without proof (since my multi-dimensional imager is not working at present) that for each such pattern in n dimensions, one can repeat and shift it so that each point is shared by at most (n+2) bricks of length n+1. Simply look at the points shared by most bricks in n dimensions, thicken to the next dimension, then place a brick squarely on top of such a point. It should be apparent that the layer can be shifted so that no point is shared by more than 1 more brick when a dimension is added. If the distance constraint needs larger bricks, scale the sides as needed.
At worst, this idea if wrong will give Mark an opportunity to clarify the situation.
EDIT: Now I understand the problem better. The difficulty with the above is that the constraint of each point on at most n+1 bricks when applied to the unit cube forces the offset to be smaller (1/2^n) as the dimension increases. That suggests to me that the point might need to be different distances from disjoint bricks in each dimension, which in turn suggests a quadratic lower bound for s(n). I will leave this here and update it with any better ideas I obtain. END EDIT.
Gerhard "Ask Me About System Design" Paseman, 2012.07.12