Circumstances Singapore Airlines Flight 222 [SIA 222] had departed Sydney for Singapore at 1548 hours EST and passed over Darwin at Flight Level [FL] 370. As this level was non-standard and the aircraft was going to enter Indonesian airspace, the crew were asked if they could accept climb to FL390 [a standard level] prior to the boundary position [KIKEM]. When the crew replied in the affirmative, Darwin ATC issued a clearance for the aircraft to climb to FL390 with a requirement to reach that level by KIKEM. This instruction was issued at 1918 hours CST and was clearly and correctly read back by the crew. Darwin ATC coordinated this information with Bali ATC as per the coordination agreement between Australia and Indonesia. At 1935 hours SIA 222 reported at the position prior to the boundary [SATKO] at FL370. No mention of FL390 was made at this time by either the crew, or ATS. Perth FS instructed the crew to contact Bali control at KIKEM. Because ATS did not mention or query the crew with respect to FL390, the Captain of SIA 222 became unsure of the validity of the prior clearance and instruction. The flight crew did not question ATS about the clearance. SIA 222 reported at KIKEM at 1956 hours but the aircraft was still at FL370 in contravention of the ATC clearance received. Bali control then coordinated with Darwin ATC to confirm the level of SIA 222 and the aircraft was subsequently recleared to FL390. There was no breakdown in separation. Significant Factors 1.The crew of SIA 222 became unclear as to the validity of their air traffic control clearance after reporting at SATKO at FL370. 2.The crew of SIA 222 did not make any radio call to clarify the validity, or otherwise, of their air traffic control clearance. 3.The crew of SIA 222 did not reach FL390 by KIKEM which was in contravention of their air traffic control clearance.