READ BACK CLRNC ON INTERPHONE MEANT FOR ANOTHER ACFT. POTENTIAL CONFLICT RESULT OF CLIMB THROUGH FL310.

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DURING CLBOUT OF CVG; F/E FOUND IT URGENTLY NECESSARY TO LEAVE COCKPIT TO VISIT TOILET. F/A'S CALLED COCKPIT AND STATED THAT THE CABIN WAS TOO HOT. F/O LEFT SEAT TO ADJUST CABIN TEMP. WHILE F/O WAS OCCUPIED AT S/O PANEL; CAPT HEARD ATC CLRNC TO FL350. CAPT ACKNOWLEDGED CLRNC TO IND CENTER. AFTER F/O RESUMED SEAT CAPT HAD F/O RESET ALT REMINDER TO FL350. CLBING THROUGH FL310; IND CENTER INDICATED CLRNC LIMIT HAD BEEN FL290 AND FLT HAD PASSED 5.3 NM FROM BIZJET AT FL310. PROB OCCURRED BECAUSE CAPT HAD TAKEN CLRNC FOR XYZ TO FL350; AND ACKNOWLEDGED IT BACK TO CENTER WITH XMITTER SELECTED TO GND INTERPHONE (FROM GATE PUSHBACK). XYZ WAS THE CALL SIGN USED ON PREVIOUS FLT INTO CVG AND DEPARTED JUST AHEAD OF US OFF CVG. FL350 WAS THE FILED ALT OF OUR FLT. SUGGEST THE PRACTICE OF HAVING AIRCREWS DEPARTING HUB CITIES AT THE SAME TIME OF THEIR PREVIOUS FLT CALL SIGN BE EVALUATED FOR SUSCEPTIBILITY OF CREWS TO ACCEPT WRONG FLT CLRNC. ALSO SUGGEST THAT ATC NEEDS TO BE ISSUING INSTRUCTIONS VIA DATALINK WITH VOICE BACKUP.