ZK-NCJ, Auckland to Brisbane, at FL350 estimated position UDIKO at 0921 EST. ZK-NBI, Brisbane to Christchurch, was on climb to FL370 on reciprocal track (A464) estimating UDIKO at 0854. The Oceanic Controller calculated a time of passing at 0907 and on this basis required ZK-NBI to reach FL370 by 0855. The inbound aircraft revised the estimate for UDIKO and the controller revised the time of passing to 0906. The controller monitored the climb of ZK-NBI and became concerned that the requirement would not be met. He asked the crew to confirm that they would reach FL370 by 0855. The reply was that they would. At 0855 the aircraft was observed to be at FL355 and this was confirmed by the crew. Traffic information was then passed to both aircraft. ANZ76 reached FL370 at 0857. FL330 was not available due to other crossing traffic. The pilot-in-command of ZK-NBI later submitted a report which said that the aircraft's Flight Management Computer gave an indication that they would be able to reach the level by the required time. When they approached within a thousand feet of the level, the aircraft performance dropped off sharply. They then asked for FL350 which was not available. The aircraft suffered an unexplained loss of performance which was not forecast by the Flight Management Computer. The reason for the loss of performance was not determined.