  Given a distribution of pebbles on the vertices of a graph, say that we can
pebble a vertex if a pebble is left on it after some sequence of moves, each of
which takes two pebbles from some vertex and places one on an adjacent vertex.
A distribution is solvable if all vertices are pebblable; the pebbling
threshold of a sequence of graphs is, roughly speaking, the total number of
pebbles for which random distributions with that number of pebbles on a graph
in the sequence change from being almost never solvable to being almost always
solvable. We show that any sequence of connected graphs with strictly
increasing orders always has some pebbling threshold which is
$\Omega(\sqrt{n})$ and $O(2^{\sqrt{2 \log_2 n}} n/\sqrt{\log_2 n})$, and that
it is possible to construct such a sequence of connected graphs which has any
desired pebbling threshold between these bounds. (Here, $n$ is the order of a
graph in the sequence.) It follows that the sequence of paths, which, improving
earlier estimates, we show has pebbling threshold $\Theta(2^{\sqrt{\log_2 n}}
n/\sqrt{\log_2 n})$, does not have the greatest possible pebbling threshold.
