Or you might actually get approved after a few days, like both me and P2 for the business card. Just don't expect a rational ordering of communication - here's what we got:
Day 0: Applied
Day 0: "Your application has been received" email [No instant approval ]
Day 6: "Thanks for going paperless" email [with no acknowledgement of approval, but I considered this a very good sign!]
Day 15: "Your new cardmember companion travel discount is now available" email [still no acknowledgement of approval, but I'm very confident at this point]
Day 21: "Welcome to the card that means business" email [ah, finally! ]