  There has been increased recent interest in understanding the relationship
between the symbolic powers of an ideal and the geometric properties of the
corresponding variety. While a number of results are available for the
two-dimensional case, the higher-dimensional case is largely unexplored. In the
present paper we study a natural conjecture arising from a result by Bocci and
Chiantini. As a first step towards understanding the higher-dimensional
picture, we show that this conjecture is true in dimension three. Also, we
provide examples showing that the hypotheses of the conjecture may not be
weakened.
