A POTENTIAL CONFLICT WAS CREATED FOR ATC WHEN THE FLC OF ONE ACR B757 TURNED INTO THE PATH OF THE SECOND B757. THE LEADING ACFT WAS IN A CLB THROUGH OCCUPIED ALT OF THE RPTR'S FLT; 20 MI NE OF SJT; TX.

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DUE TO WX; WE REQUESTED A RERTE FROM PRESENT POS TO SAN ANGELO (SJT) WHICH WAS GRANTED BY ATC. FOLLOWING SJT; WE WERE CLRED DIRECT CNX; FLT PLAN RTE. WE FURTHER REQUESTED AND RECEIVED CLRNC TO DEVIATE DUE TO WX. TO AVOID ADDITIONAL WX AT SJT; WE TURNED TOWARDS CNX WHEN WE WERE APPROX 20 MI FROM SJT. WE WERE WORKING ATC FREQ 132.07 AT THIS TIME AND COULD HEAR CTR TALKING TO AN ACR FLT. ACR WANTED TO DEVIATE AND ATC CLRED HIM FROM FL370 TO FL410 AND INSTRUCTED HIM 'NOT TO DELAY THE CLB.' 6 MINS LATER; WE OBSERVED A CONTRAIL NEAR 12 O'CLOCK POS THAT APPEARED TO BE TRAVELING IN OUR SAME DIRECTION BUT LOWER. THEN WE RECEIVED A TCASII WARNING AND HEARD ATC INSTRUCT THE FLT TO TURN L AND TO IMMEDIATELY CLB TO HIS ASSIGNED ALT OF FL410. WE OBSERVED THE OTHER JET AT THE 11:45 POS AND HEARD ATC REQUEST THAT WE TURN L ALSO. THE AUTOPLT WAS DISCONNECTED; A L TURN WAS ESTABLISHED; THE OTHER JET CROSSED THE NOSE OF OUR FLT INDICATING FL392 AND PASSED TO OUR R. WE ESTIMATE THAT OUR 2 JETS PASSED WITHIN 1/2 MI HORIZLY; AND 200 FT VERTLY. FOLLOWING THE NEAR MISS; ATC INDICATED TO OTHER FLT THAT HE HAD NOT BEEN CLRED OR EXPECTED TO TURN THAT FAR S AND THAT HE WAS SUPPOSED TO HAVE CLBED WITHOUT DELAY TO FL410. CONTRIBUTING FACTORS: 1) WX. 2) OTHER FLT NOT FOLLOWING ATC DIRECTIONS TO 'CLB WITHOUT DELAY.' 3) OTHER FLT TURNING S WHEN NOT EXPECTED BY ATC. NOTE: OTHER FLT EVASIVE ACTION SMOOTH ENOUGH THAT PAX AND FLT ATTENDANT DID NOT NOTICE. 1 PAX AND 1 FLT ATTENDANT DID SEE THE OTHER JET FLY BY TO OUR R (FROM PAX WINDOW).