Air carrier Captain in ZJX airspace reported confusion about a possible erroneous altitude clearance received via CPDLC which resulted in an altitude excursion.

Narrative

Midway into the flight we requested a climb to FL360 via CPDLC. We received a Standby" response. We acknowledged the standby and remained at FL340. A few minutes later we saw the flashing blue incoming message on the glareshield. We both read "CLB (Climb) & Maintain FL360". The first officer pressed 'confirm" and we proceeded to climb. Around FL34.5; ATC called and told us to stop our climb at FL350. There were no TCAS alerts. ATC did not issue any urgent commands to any aircraft on our frequency. We saw no traffic nearby. ATC asked us about receiving a "standby" response on our request to climb. We explained that we did initially receive the standby response and that we had received a clearance to climb to FL360 afterward. We leveled at FL350 for a few minutes. Then ATC cleared us up to FL360. We were handed off to the next sector. It seems that our initial request to climb to FL360 must have re-appeared or 'bounced back' after the standby message from ATC. We looked in the message record and all we saw was our initial request to climb and the Standby response back from ATC. There was no message telling us to climb to FL360. We may have been mistaken in reading incoming message. When the message bounced back it is possible we both misread it as a clearance to climb rather than our initial request. We do not think we were in error as neither one of us saw the word 'Request' in the incoming message.About ten minutes after this ATC notified us that they could not clear our climb request from their system and asked us if we could. We were unable. They then let us know they had been able to get it cancelled in their system.A very similar situation happened the next week in the same airspace. I received a 'bounced back' message 30 minutes after my initial request. However this message said clearly at the top "Request FL xxx." There was no doubt this message was a 'bounce back'. Unfortunately at the time I did not think to obtain a copy. We ignored the message and proceeded as planned. The first officer and I were very careful to confirm with each other when receiving clearances via CPDLC. This event seemed out of the ordinary. It seemed to be a system error. Moving forward I will read the clearances more carefully and I will use direct voice contact for climb requests or any requests that are slightly out of the ordinary."