A C560XL CREW IS UNABLE TO CLB TO A CLRED ALT OF FL450 AS REQUESTED TO CLR OPPOSITE DIRECTION TFC. A CONFLICT RESULTS.

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WE WERE LEVEL AT FL430 EBOUND. ATC ASKED US TO EITHER CLB TO FL450 OR DSND TO FL410 DUE TO TFC COMING OUR WAY AT FL430. PRIOR TO THIS; WE HAD DETERMINED THAT FOR OUR WT AND THE OUTSIDE AIR TEMP; WE COULD BE ABLE TO CLB TO FL450 IF NEEDED. I (CAPT) WAS FILLING OUT PAPERWORK AND PROGRAMMING THE FMS. THE COPLT; WHO WAS THE PF; STATED THAT WE WOULD CLB TO FL450. AT ABOUT FL446; THE ACFT WAS UNABLE TO CLB ANY FURTHER. WE INFORMED ATC THAT WE WOULD NEED A DSCNT AND WOULD BE UNABLE TO STAY WHERE WE WERE. ATC INSTRUCTED US TO DSND TO FL410 AND TO TURN 40 DEGS L. WHILE DSNDING; WE APPARENTLY CAME WITHIN 1.5 MI AND 1200 FT OF THE TFC AT FL430. WE HAD THE OTHER ACFT INSIGHT; AND HE ALSO STATED THAT HE HAD US IN SIGHT. WE RECEIVED A TA ON THE TCASII AS WE WERE MANEUVERING. ATC LATER GAVE US A NUMBER TO CALL AND INFORMED US OF A POSSIBLE PLTDEV. AT NO TIME DID I FEEL THE SAFETY OF FLT WAS COMPROMISED. I FEEL THAT I SHOULD HAVE MONITORED MORE WHAT THE OTHER PLT WAS DOING; AND WE SHOULD HAVE PAID MORE ATTN TO THE TEMP AS WE CLBED; AS I FEEL THAT WAS THE DETERMINING FACTOR IN OUR INABILITY TO CLB ANY FURTHER.