C172 PLT MAY HAVE HAD A RWY INCURSION ON RWY 15 AT OAK.

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THIS WAS A CHKOUT AND FAMILIARIZATION FLT FOR ME. I HAD PREVIOUSLY FLOWN REGULARLY AT THE OAKLAND ARPT; BUT NOT FOR SEVERAL YRS. AT THE COMPLETION OF THE FLT; THE ACFT WAS FUELED AT THE FBO AND WAS BEING TAXIED TO THE 'OLD T HANGARS' ON THE N FIELD. TAXI CLRNC WAS 'TAXI TO OLD T'S; HOLD SHORT OF RWY 15; ADVISE HOLDING SHORT.' THE RTE WAS TXWY D; CROSS RWY 15 AT THE APCH END; TXWY N. UPON ARRIVING AT RWY 15; WE HELD SHORT AND CALLED GND. (I BELIEVE) GND CLRED US ACROSS RWY 15 AND I READ BACK THAT XING CLRNC. AS WE CROSSED THE RWY; A CALL WAS MADE TO ANOTHER ACFT; WHICH I THOUGHT WAS FOR US; SAYING SOMETHING ABOUT TXWY A (LOCATED AT THE OTHER END OF THE ARPT). I CALLED THE GND CTLR AND ASKED GND ABOUT USING TXWY A. NO RESPONSE CAME. AFTER CLRING THE RWY; I CALLED GND AGAIN ASKING THEIR REQUEST FOR US REGARDING TXWY A. GND CAME BACK AND SAID THAT THEY WERE TALKING TO SOMEONE ELSE (NOT US) AND ASKED WHAT WE NEEDED. I RESPONDED THAT I THOUGHT THEY WERE TALKING TO US; MY MISTAKE; WE DIDN'T HAVE ANY OTHER REQUESTS. THE CHIEF PLT OF THE CLUB (WHO WAS CONDUCTING THE CHKOUT) THOUGHT WE MAY HAVE CROSSED RWY 15 WITHOUT A CLRNC. I TOLD HIM THAT WE HAD BEEN GIVEN A CLRNC WHICH I HAD READ BACK AND THAT THE CONFUSED EXCHANGE WITH THE GND CTLR TOOK PLACE AFTER THAT CLRNC WAS RECEIVED. THERE IS NOW SOME DOUBT IN EACH OF OUR MINDS ABOUT WHAT TRANSPIRED.