A DC9 CREW MISINTERPED CLRNC; PROCEEDED DIRECT; BEFORE ARTCC INTENDED.

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FLT XXX DEPARTED PHL FOR BNA ON 01/01/THUR. OUR DEP FROM PHL AND CLB UP TO FL290 WAS WITHOUT INCIDENT. AT FL290; WE WERE IN CONTACT WITH ZNY. WE WERE FLYING WBOUND ON J110 AND WERE ABOUT TO INTERCEPT J6 ON OUR FLT PLAN RTE WHEN ZNY ASSIGNED US A HDG OF 260 DEGS AND TO CLB TO FL350 AT 300 KIAS OR GREATER. WE DID REQUEST A MACH SPD FOR THE CLB. WE WERE TO CLB TO FL330 AND THEN RESUME NORMAL SPD; TO PROCEED DIRECT TO MARTINSBURG; JOIN J6 AND RESUME OUR OWN NAV. WE TURNED TO MARTINSBURG NAVAID AND OUT OF FL310 NOTICED ON OUR TCASII AN ACFT OFF OUR L WING ABOUT 5 NM AND 300 FT BELOW US AS WE CONTINUED OUR CLB. NO TA OR RA APPEARED ON THE TCASII. WE IMMEDIATELY TURNED R IN A 30 DEG ANGLE BANK TO A HDG OF 240 DEGS; AT WHICH TIME ATC ASKED US WHAT OUR HDG WAS AND TOLD US TO TURN FURTHER R TO A HDG OF 280 DEGS. WE CONTINUED ON OUR ASSIGNED HG; CONTINUING OUR CLB TO FL350. WE DID NOT HEAR XMISSIONS FROM THE OTHER ACFT AT FL310. THE REMAINDER OF THE FLT TO BNA WAS WITHOUT INCIDENT. I BELIEVE WHAT CAUSED THE SIT OF 'LOSS OF SEPARATION' BTWN ACFT WAS THE CTLR'S INTENTION IN THE ISSUED CLRNC AND OUR INTERP OF THE CLRNC; A MULTIPLE CLRNC. WITHOUT A DOUBT; WE SHOULD HAVE CLARIFIED THE GIVEN CLRNC; VERIFYING WITH THE CTLR OUR UNDERSTANDING OF THE SAID CLRNC.