Data point:
I called 1-888-245-0625 to check on the status of my app. and was told I was declined because I'd recently been issued a BA card (November, 09). I asked for a super. and he said that if I closed one of my 3 Chase cards (BA, SWA and and a Gold UA Visa that only earns 1 mile per $2) he'd approve me. But all my cards currently have balances so he couldn't do it now. He said "call back after you've paid off a card & canceled it". Hope this works.