I forgot the most important part. I'm currently paying a judgement for this debt. They won the judgement a long time ago and I never knew about this until may of this year when they began to garnish my wages. The judgement is not on my credit report. So I would say it is safe to assume they do not blacklist.
Thanks for the responses. I have a paid judgement on my credit report (with Capital one though). US Bank and BOA were kind enough to open accounts for me, although they asked me to about the judgement. They were ok with it as it was paid.
I guess the only way to find out about Chase is to apply and see what happens. I have one hard inquiry falling off my report at the end of this month. I will check what my scores are like and then decide if I should go for it.