Often, yes, especially local credit unions. They aren't part of huge bank chains, so they have more flexibility in determining eligibility. Some of the big national ones, like Penfed, can be tougher. Also, CU's often give out higher CL's, even for starters and starting-overs.
If you're interested in a CU, or even a local non-chain bank, if such a thing exists any more, I would pull a fresh FICO EQ, print out the entire score and score report, and take it with you while you visit local institutions. Ask to speak to a loan officer, tell them that you're not yet authorizing a hard inq and that you know that this would be unofficial, but ask if they would take a look at your EQ report and give their opinion as to what you might be eligible for. Once you find a place you like, you can go ahead and do a formal app.
The CU's in my area have banded together to provide free ATM service at any participating CU, which is handy for pulling out some cash when you need it without having half of it taken away by fees. My own local CU also participates in some sort of national rewards points clearinghouse. (I don't have a CC from them, only banking, so I don't know the details.)
note: the majority of CU's use EQ. Patelco on the West Coast uses EX, though.
* Credit is a wonderful servant, but a terrible master. * Who's the boss --you or your credit?
FICO's: EQ 781 - TU 793 - EX 779 (from PSECU) - Done credit hunting; having fun with credit gardening. - EQ 590 on 5/14/2007