MLG OVERSHOT ALT INTENDED BY ARTCC AND HAD LESS THAN STANDARD SEPARATION WITH ANOTHER MLG.

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CLBING OUT OF DTW; ENTRE IAD; BEING VECTORED FOR V-103 TO ACO VOR. F/O FLYING;AND APPROX FL210 (CLRED TO FL240); I DECIDED TO TURN OFF THE SEAT BELT SIGN. I STATED TO THE F/O I WAS 'GOING OFF THE AIR' TO MAKE THE COMPANY REQUIRED P.A. ANNOUNCEMENT ABOUT THE SEAT BELT; THEN TURNED DOWN MY VHF VOLUME TO DO SO. WHILE TALKING ON THE P.A. I NOTICED AND POINTED TO CONVERGING TFC ABOVE US AND TO OUR R. I CHKED OUR ALT (JUST ABOVE FL240) AND THE ALT ALERT SELECTOR (NOW SET AT 26000'). JUST AS I FINISHED THE ANNOUNCEMENT AND BROUGHT MY VHF VOLUME BACK UP; A F/A KNOCKED ON THE DOOR AND REQUESTED OUR ETA AND ASKED US TO 'WARM IT UP IN BACK.' I GAVE HIM THE ETA WHILE THE F/O ADJUSTED THE TEMPERATURE. IMMEDIATELY THEREAFTER; AS WE WERE PASSING APPROX 25300'; CLEVELAND CENTER GAVE US A FREQ CHANGE. I ACKNOWLEDGED THE CHANGE; THEN SAID; 'BEFORE WE GO; WHAT'S THE ALT OF THAT TFC AT OUR 1 O'CLOCK?' CENTER SAID; 'HE'S AT 260; WHAT'S YOUR ALT? YOU'RE ONLY CLRED TO 250.' (WORDS APPROX). WE IMMEDIATELY STOPPED OUR CLB AT ABOUT 25800' AND DSNDED TO FL250; PASSING APPROX 1-2 MI BEHIND AND BELOW AN AIRLINES MLG. WHILE WE WERE DOING THIS; I TOLD CENTER I BELIEVED WE WERE CLRED TO AND HAD ACKNOWLEDGED FL260. IN RETROSPECT; THIS WAS AN ASSUMPTION ON MY PART; SINCE THAT CLRNC WAS ISSUED AND ACKNOWLEDGED WHILE I WAS OFF THE AIR MAKING THE ANNOUNCEMENT. THE F/O LATER SAID HE WAS FAIRLY CERTAIN THAT HE HAD RECEIVED FL260. SINCE HE IS A VERY ALERT AND PRECISE F/O; I HAVE NO REASON TO BELIEVE OTHERWISE. AT THE TIME OF THIS WRITING; WE HAVE NO KNOWLEDGE OF WHAT IS ON THE ATC TAPES; BUT IT'S OBVIOUS THAT ONE OF THREE SCENARIOS OCCURRED: 1) CLEVELAND CENTER ISSUED AN IMPROPER CLRNC TO FL260; OR 2) CENTER ISSUED A CLRNC TO FL250 BUT THE F/O MISUNDERSTOOD; ACKNOWLEDGED; AND SET FL260; OR 3) FL250 WAS ISSUED AND ACKNOWLEDGED; BUT THE ALT ALERT WAS MIS-SET TO 26000'; AND THE ERROR WAS NOT CAUGHT; POSSIBLY DUE TO THE SUBSEQUENT DISTRS. IF SCENARIO 1 IS THE CASE; IT WAS AN ERROR BY CENTER THAT WE IN THE COCKPIT HAVE NO CONTROL OVER. IF 2 IS TRUE; THE RESPONSIBILITY IS SHARED BY US AND ATC; SINCE CENTER DID NOT NOTICE THE DISCREPANCY DURING THE READBACK. IN THE CASE OF 3; WE ALONE IN THE COCKPIT ARE RESPONSIBLE. IN EITHER CASE 2 OR 3; THE CHANCE FOR ERROR WOULD HAVE BEEN GREATLY REDUCED HAD I BEEN 'IN THE LOOP' 100% OF THE TIME. HOWEVER; WITH A TWO PLT CREW; THIS IS NOT ALWAYS POSSIBLE (ATIS; MAINT PROBS; REQUIRED PA ANNOUNCEMENTS; ENG READINGS; ETC). WE'VE ATTEMPTED TO REDUCE THE IMPACT OF THESE 'EXTRA' DUTIES BY THE STERILE COCKPIT CONCEPT BELOW 10000'; BUT UNLESS WE RETURN TO THREE PLT CREWS OR USE MANY NEW AUTOMATIC DEVICES (TAPED ANNOUNCEMENTS; UPLINK TO COCKPIT PRINTER FOR ATIS; AUTOMATIC ACARS ENG READINGS; ETC) WE HAVE TO ACCEPT THE OCCASIONAL PERIODS OF ONE PLT BEING 'SOLO'. I DON'T PARTICULARLY LIKE THIS; BUT THEN I DON'T KNOW WHAT TO DO ABOUT IT EITHER.