Same thing happened to me at SFO a few weeks ago before an AY J flight. Was told the lounge was full (which of course one can't tell from the entrance), although it was an hour or two before the A380 (BA286) departure so I assumed that was a fairly accurate statement.
The attendant told me to go to the AF/KL lounge that AY uses for their J passengers. Went over to the CX lounge instead and got in there without any problem.