TOTALLY YMMV, but here was my experience. I hadn't checked for prequals, but just a US Bank card experience.
I wanted a Cash+ but had just gotten 4 cards (between primary and au). I opened a checking account with them (was part of applying for a heloc, but I don't think that mattered). As part of the whole walk-through in branch of opening checking, there was a list of CC offers. I got my Cash+. I don't know if they're still offering $125 for opening a checking (it was only $75 when I got it) but they seem to be pretty relationship driven and if you forsee wanting to do business with them, I'd go for a checking account. If you have a CC with them, the monthly fees are waived on a gold checking (at least in my area). IDK if it mattered, but it was opened as a platinum package (first two months free) and I'm having it down graded to gold...platinum has fees waived with heloc, CC only waives fees on gold.
They've also got a savings thing where if you 'get' $1000 into your savings account, they give you a $50 giftcard. Then if you leave it there for 1 year, you get another. You have to have it transfered there via a 'savings goal'. So I had $25 transferred the first month, but when I was in a position to park $1000 there, I transferred the other $975. I just got my first $50 and will wait for my second. I'm just spreading out my emergency cash. (letting DCU give me 3% on $500 too).
So US Bank may have been ultra-low on what they valued my house on preventing the heloc, but I got my Cash+ and have made $125 off of them so far. And the branch manager loves us. He will continue to court us, I think. 
He knows they blew it on the appraisal.
