ACR X HAD LTSS FROM ACR Y. SYS ERROR.

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ACR X PHL TO EGKK CLBING TO FL270. ACR Y BOS TO MIA FL310. ACR Z YHZ TO CHS AT FL280 SWBOUND J174. SECTOR HAD BEEN VERY BUSY AND ALL BUT 6 ACFT GONE. ACR X WAS CLBING TO FL270 STOPPED FOR ACR Z SBOUND AT FL280. CTLR ASKED ACR X FOR RATE OF CLB. PLT RPTED 1500 FPM. CTLR RAN OUT VECTOR LINES AND DETERMINED THAT HE COULD TOP ACR Z AND TOLD ACR X TO CLB TO FL370; EXPEDITE THROUGH FL290. CTLR ASKED AGAIN WHAT ACR X RATE OF CLB WOULD BE; AND PLT AGAIN RPTED 1500 FPM. CTLR DETERMINED THAT THIS CLB WOULD ALLOW ACR X TO CLB AND NOT BE A FACTOR WITH ACR Y AT FL310 ABOUT 15 MI AHEAD OF ACR Z SBOUND ON J174. ACR X CLBED AT 3500 FPM BECOMING IN CONFLICT WITH ACR Y AT 310. WHEN CTLR NOTICED CONFLICT HE ATTEMPTED TO STOP ACR X WHO WAS ALREADY TO FL296 AND 2.5 MI FROM ACR Y. SUPPLEMENTAL INFO FROM ACN 203056. ACR X DEPARTING PHILADELPHIA FOR GATWICK WAS STOPPED IN THE CLB AT FL270 FOR XING TFC AT FL280. ACR Z (HALIFAX TO CHARLESTON). AT ONE POINT; THE RADAR CTLR DECIDED THAT ACR X WAS CLBING WELL ENOUGH TO TOP ACR Z; AND ASKED ACR X IF HE COULD MAINTAIN 1500 FT OR BETTER THROUGH FL290. THE ACFT SAID YES AND WAS THEN CLRED TO FL370. A SHORT TIME LATER; THE CTLR FOUND IT NECESSARY TO TURN BOTH ACFT TO THE R TO KEEP SEPARATION. IN THIS; HE WAS SUCCESSFUL. HOWEVER; HE DID NOT NOTICE ANOTHER ACFT AT FL310. ACR Y; BOSTON TO MIAMI. I WAS WORKING THE RADAR ASSOCIATE POS AND DID NOT NOTICE IT EITHER. THE CONFLICT ALERT WAS TRIGGERED AND THE CTLR STOPPED ACR X AT FL290 BUT UNFORTUNATELY THE PLANE WAS CLBING SO WELL THAT HE WAS NOT ABLE TO AVOID GOING THROUGH BALTIMORE. ACR X LEVELED AT FL296 BEFORE STARTING BACK DOWN TO FL290. THE OPERROR DEVICE DETECTED THE ERROR. I SIGNED ON AT THE RADAR ASSOCIATE POS AT THE SECTOR ADJACENT TO THIS ONE. AFTER A WHILE THE RADAR CTLR SUGGESTED THAT I MOVE OVER TO THE NEXT SECTOR (HAMPTON) TO HELP THE RADAR CTLR THERE WHO WAS GETTING QUITE BUSY. THIS LEFT MY PREVIOUS SECTOR STAFFED BY ONE CTLR WHO WAS QUALIFIED AT THAT SECTOR; BUT NOT FULLY QUALIFIED AT ALL SECTORS IN THE AREA. (THIS PRACTICE IS AGAINST FACILITY POLICY). WHILE AT THE HAMPTON SECTOR; THE CTLR THERE WAS RELIEVED BY THE ONE WHO WOULD LATER HAVE THE ERROR. THE TFC WAS NOT PARTICULARLY HVY AT THE TIME OF THE INCIDENT.