  In a nesting of a balanced incomplete block design (or BIBD), we wish to add
a point (the \emph{nested point}) to every block of a $(v,k,\lambda)$-BIBD in
such a way that we end up with a partial $(v,k+1,\lambda+1)$-BIBD. In the case
where the partial $(v,k+1,\lambda+1)$-BIBD is in fact a
$(v,k+1,\lambda+1)$-BIBD, we have a \emph{perfect nesting}. We show that a
nesting is perfect if and only if $k = 2 \lambda + 1$.
  Perfect nestings were previously known to exist in the case of Steiner triple
systems (i.e., $(v,3,1)$-BIBDs) when $v \equiv 1 \bmod 6$, as well as for some
symmetric BIBDs. Here we study nestings of $(v,4,1)$-BIBDs, which are not
perfect nestings. We prove that there is a nested $(v,4,1)$-BIBD if and only if
$v \equiv 1 \text{ or } 4 \bmod 12$, $v \geq 13$. This is accomplished by a
variety of direct and recursive constructions.
