A consequence of the Cabling Conjecture of Gonzalez-Acu\~{n}a and Short is
that Dehn surgery on a knot in $S^3$ cannot produce a manifold with more than
two connected summands. In the event that some Dehn surgery produces a manifold
with three or more connected summands, then the surgery parameter is bounded in
terms of the bridge number by a result of Sayari. Here this bound is sharpened,
providing further evidence in favour of the Cabling Conjecture.