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Roughly what kind of credit limit are you looking for? Different people have different perceptions of what a high limit is.
There was a long stretch where USBank was notorious for approving $500 Cash+ Visa Signatures (the 2 zeros are correct) so don't get yourself down over their giving you a $1200 starting limit.
^^ Thanks for the reply, I was about to edit my post because I realize that what I'm asking isn't actually a very "high" credit limit - I'm looking for around $4-5k. It's more that I just don't want to get a disgustingly low CL (i.e. like the US Bank $1200). I don't want to waste hard inquiries if I know the particular banks are notorious for issuing low lines.