You brought back two pleasant memories and I can confirm the First USA, the issuer of the United card, does allow multiple accounts and bonuses. I got rejected for a British Airways card, which made me mad, so I applied again and got rejected again and got mad again. I hadn't realized that First USA was owned by Bank One, where I had a relatively large but unused line of credit. I closed the line of credit and asked First USA to review my application for a British Airways card. They approved both applications and I got two 15,000 mile bonuses.

I am also a sucker for the cool Southwest Airlines shirts they give away for free at PHX when you apply for a card. I figured I would get the shirt and simply be rejected for the card, since I already had a S/W Visa card. I wasn't rejected and I ended up with three S/W Visa cards and two S/W Airlines T-Shirts. Best of all both of my additional S/W Visa cards generated the 2 Bonus Credits on first use and the five bonus credits for balance transfers.