Also as an FYI - You might be able to shuffle credit lines around with Barclay or Citi so they are not giving you new credit, and lower your credit lines on open cards to let you open another. For example if you had 3 cards open at $10k limits each, maybe they will let you lower the limits on the open cards to allow you to open the AA card. 4 cards at $7500 limits is not really any different to Barclay's from a risk perspective than 3 at $10k.

I've also had a case (with Citi I believe) where the internal guideline was only let you open the card if they could give you a $10k limit for that particular card, and the computer denied it because they only wanted to give me $9600. When I called reconsideration and they explained it, I said I would be fine with a $9600 limit and they pushed it through.

I know chase has a hard and fast 5 cards from anyone in 24 months rule, but as far as I know Barclays does not.