I've noticed that most of the CX business fares now only allow a stopover in one direction, and also limit that stopover to 7 days now. As such you may want to try playing about a bit with the dates you are trying to do the stopover on.
An example that is fairly typical on an intra-asia flight between cities:
Originally Posted by expertflyer
UNLIMITED STOPOVERS PERMITTED ON THE OUTBOUND AND ON
THE INBOUND FARE COMPONENTS
NONE IN HKG OUTBOUND.
A STOPOVER MAY NOT EXCEED 168 HOURS.