FLT CREW CONFUSION AS TO ALT ASSIGNMENT.

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WE WERE CRUISING AT FL270 THE F/O LEFT THE COCKPIT TO GO AFT TO THE CABIN. A FEW MOMENTS LATER ATC CLRED US TO CLB TO WHAT I BELIEVE WAS FL290. THE CAPT ACKNOWLEDGED THE CLB CLRNC AND ALT AND SET 29;000' IN THE ALT SELECT WINDOW. TFC WAS RPTED AT OUR 8 O'CLOCK POS; WHICH WE DID NOT SEE AT THE TIME. APPROX 1 MIN LATER; THE CAPT SPOTTED THE TFC AND REMARKED THAT IT APPEARED TO BE CLOSER IN DISTANCE AND ALT THAN EXPECTED. THE CAPT THEN ASKED ME IF I AGREED THAT OUR CLRED ALT WAS FL290 AND I ANSWERED YES. WE WERE CLBING THROUGH ABOUT FL280 WHEN THE F/O RETURNED TO THE COCKPIT. THE CAPT THEN ADVISED THE F/O THAT WE WERE CLBING TO FL290. CLBING THROUGH FL284; ATC CALLED FOR US TO MAINTAIN FL280. THE F/O ACKNOWLEDGED THE CLRNC; ADVISING CENTER WE WERE DSNDING FROM FL284 TO FL280. CENTER THEN ADVISED US THAT THE CLRED ALT HAD BEEN FOR FL280 AND OF A POSSIBLE ALT DEVIATION RPT. AT NO TIME WAS THERE A QUESTION IN MY MIND ABOUT THE ALT WE HAD BEEN CLRED TO OR THAT THE CAPT HAD ACKNOWLEDGED A CLB TO FL290. THE CAPT CONFIRMED THE ALT WITH ME ONLY BECAUSE OF THE PROX OF THE OTHER ACFT WHICH PASSED BEHIND US AND AT NO TIME WAS THERE A HAZARDOUS SITUATION. NO DEVIATION FROM STANDARD RADIO AND OPERATIONAL PROC OCCURRED. AFTER TALKING WITH ATC; THE CAPT SAID THAT THE TAPES SHOWED THAT WE HAD BEEN CLRED TO FL280 AND THAT HE HAD ACKNOWLEDGED THE CLRNC TO THAT ALT. THE MISUNDERSTANDING ON MY PART MAY HAVE BEEN CAUSED BY THE ANTICIPATION OF A CLRNC TO FL290; BECAUSE THAT WAS OUR FILED FINAL ALT. THE SITUATION COULD PROBABLY HAVE BEEN AVOIDED HAD WE CONFIRMED THE ALT AFTER SPOTTING THE OTHER ACFT WHEN IT SEEMED THAT IT WOULD PASS CLOSER THAN NORMAL SEP. SUPPLEMENTAL INFO FROM ACN 92250: I DO NOT KNOW HOW THE CLRNC TO FL290 CAME ABOUT; HOWEVER; THE READBACKS AND PROCEDURAL REPORTING; WHICH INCLUDED THE CLRNC TO FL290; DID NOT HAVE THE EFFECT OF ALERTING THE 2 CTLRS THAT WE WERE CLBING TO 290 (IF THAT WAS NOT THEIR INTENT). THE IND ARTCC CTLR WAS CLEARLY INVOLVED IN THE CTL OF A NUMBER OF OTHER ACFT; SOME OF WHICH; AS I RECALL; WE DID NOT HEAR ACKNOWLEDGING XMISSIONS; LEADING TO MY SPECULATION THAT THEY WERE ON ANOTHER FREQ.