  Fractal nests are sets defined as unions of unit $n$-spheres scaled by a
sequence of $k^{-\alpha}$ for some $\alpha>0$. In this article we generalise
the concept to subsets of such spheres and find the formulas for their box
counting dimensions. We introduce some novel classes of parameterised fractal
nests and apply these results to compute the dimensions with respect to these
parameters. We also show that these dimensions can be seen numerically. These
results motivate further research that may explain the unintuitive behaviour of
box counting dimensions for nest-type fractals, and in general the class of
sets where the box-counting dimension differs from the Hausdorff dimension.
