FREQ CONGESTION AND BLOCKING RESULT IN A RWY XING WITHOUT CLRNC AT BUF; NY.

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WE WERE TAXIING FROM THE FBO WHERE WE PARK FOR THE OVERNIGHT IN BUF TO THE MAIN TERMINAL. TAXI INSTRUCTIONS WERE TO HOLD SHORT OF RWY 5 O TXWY E. AFTER WAITING A SHORT TIME FOR INSTRUCTIONS TO CROSS RWY 5; XYZ AIRLINES WAS GIVEN CLRNC TO TAXI INTO POS AND HOLD ON RWY 5. THE GND CTLR ISSUED US INSTRUCTIONS TO CROSS RWY 5. I READ BACK THE TAXI INSTRUCTIONS USING OUR COMPLETE CALL SIGN AND CROSSED THE RWY HEADED FOR THE GATE. WE CHKED L AND R PRIOR TO XING. ON OUR R; WE COULD SEE XYZ ENTERING RWY 5 AND A ZZZ ACFT XING TO THE SAME SIDE WE WERE ON. ZZZ WAS ON TXWY F AND WE WERE ON TXWY E. WHEN WE SAFELY CROSSED AT TXWY E AND STARTED DOWN TXWY A TO THE GATE; GND CTL ASKED IF THAT WAS US ON TXWY A. I CONFIRMED THAT IT WAS. THE CTLR THEN INFORMED US THAT WE HAD NOT BEEN ISSUED XING INSTRUCTIONS. HE TOLD US IT WAS FOR ZZZ. I DON'T REMEMBER THE ZZZ FLT NUMBER. I DID INFORM THE GND CTLR THAT HE HAD GIVEN US INSTRUCTIONS TO CROSS AND I READ THEM BACK TO HIM BEFORE XING. HE INFORMED US THE INSTRUCTIONS TO CROSS RWY 5 WERE FOR ZZZ AND THAT MY READBACK TO HIM WAS BLOCKED AND HE DID NOT HEAR MY RADIO XMISSION ABOUT XING. WE APOLOGIZED FOR THE MISTAKE. AS CAPT; I ASSUME ALL THE RESPONSIBILITY FOR THE IMPROPER RWY XING. FACTORS THAT I FEEL THAT CONTRIBUTED TO THIS WERE RADIO CONGESTION AND CTLR WORKLOAD. THE GND CTLR WAS WORKING BOTH GND AND TWR. EVEN THOUGH I WAS 99% SURE HE GAVE US XING INSTRUCTIONS; I WILL CONFIRM A SECOND TIME BEFORE XING; ESPECIALLY WHENEVER THERE IS A LOT OF RADIO CONGESTION. IT IS MY BELIEF THAT A CTLR SHOULD NOT HAVE TO WORK BOTH TWR AND GND FREQS; ESPECIALLY DURING BUSY TIMES.