A B777 FLT CREW THOUGHT THEY RECEIVED A CLRNC TO FL330; BUT ARTCC THOUGHT IT WAS ONLY TO FL290.

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DURING CLB AFTER LEVELING AT FL290; ZOA CLRED US TO CLB TO FL330. WE PUT FL330 IN THE ALT WINDOW; BOTH PLTS CONFIRMED IT. THE FO CHKED THE VNAV PAGE TO CONFIRM THAT OUR WT WAS COMPATIBLE WITH FL330. PF PRESSED THE BUTTON AND WE CLBED TO FL330. WE WERE GIVEN A FREQ CHANGE TO 134.450 (FREQS MAY BE REVERSED). WE CHKED IN AND AFTER A FEW SECONDS THE CTLR QUESTIONED OUR ALT. WE TOLD HIM WE WERE AT FL330 AND HAD BEEN CLRED TO GO THERE. HE SAID WE SHOULD HAVE BEEN AT FL290 AND GAVE US A TURN AND A DSCNT. WE SAID THE PREVIOUS CTLR HAD CLRED US TO FL330. HE SAID THE PREVIOUS CTLR HAD CLRED US TO FL290. AT THIS POINT THE CTLR SEEMED A LITTLE UNCOMFORTABLE AND GAVE US A CLB TO FL330. WE WERE HVY (APPROX 600000 LBS) AND THE CLB WAS UNDERSTANDABLY SLOW. HE ASKED US TO INCREASE OUR CLB RATE TWICE (I THINK). WE TOLD HIM WE WERE HVY AND WERE DOING THE BEST WE COULD. WE RETURNED TO FL330 AND THAT WAS THE END OF IT. I IMMEDIATELY SENT A MESSAGE TO DISPATCH AND ASKED THE ATC COORDINATOR TO CHK WITH OAKLAND TO SEE WHAT HAPPENED. DISPATCH SENT US A MESSAGE SAYING THAT THERE HAD BEEN NO LOSS OF SEPARATION OR ANY ALARMS. I DON'T BELIEVE THIS WAS OUR PROB. WE DIDN'T JUST DREAM UP A CLRNC TO FL330. THERE WAS NO COCKPIT CONVERSATION GOING ON AT THIS POINT; NO DISTRACTIONS WHATSOEVER. ALL PROCS WERE FOLLOWED PERFECTLY; IT'S JUST THAT BOTH OF US FEEL STRONGLY THAT WE HEARD AND READ BACK FL330.