  We study two types of nestings of balanced incomplete block designs (BIBDs).
In both types of nesting, we wish to add a point (the nested point) to every
block of a $(v,k,\lambda)$-BIBD in such a way that we end up with a partial
$(w,k+1,\lambda+1)$-BIBD for some $w \geq v$. In the case where $w > v$, we are
introducing $w-v$ new points. This is called a weak nesting. A strong nesting
satisfies the stronger property that no pair containing a new point occurs more
than once in the partial $(w,k+1,\lambda+1)$-BIBD. In both cases, the goal is
to minimize $w$. We prove lower bounds on $w$ as a function of $v$, $k$ and
$\lambda$ and we find infinite classes of $(v,2,1)$- and $(v,3,2)$-BIBDs that
have optimal nestings.
