I just called so I hope I can explain this well...
I put in my application online, and I have a not-so-high score, and the site said within 30 days I'd be notified, so I called the recon number immediately. Rings, says "Your call may be recorded" and then you get someone who will hopefully speak clearly. (Mine didn't). So I said I just applied for a card and wanted to check on the status. He said normally it takes 24 - 48 hours to get the app in the system but he'd look.
Asked for my SSN, name, address, and birthdate, and then told me that I had currently been turned down but to hang on and he'd tell me exactly why. Mine was "Seriously delinquent accounts." We got to talking about some of my previous mistakes (college screwups) and how I had one account in collections for 1600$ (DV'd, but not showing as disputed yet... gonna have to look into that). For the record, the BP Chase Visa, he was looking at my EQ report.
He seemed sufficiently pleased with my answers and started to discuss my income situation. I explained how I was working full time now and wouldn't have a problem making the payments and how I wanted to use the BP Visa as a gas card and have it as an emergency source to rebuild my credit. I figured if I'm going to be putting gas in my car, why not put the money to good use at the same time building my scores?
He said that after listening to everything I had to say, he was recommending a BP Visa with a $1000 CL (More than enough for me). While we waited for the system to decide if that was ok, we chatted about what my degree was in and what courses he took (We both started in Computer Science apparently, he moved to Business though), and soon enough he came back to me saying the system issued an authorization number, so I was indeed approved.
He then just reminded me that it's about being responsible and if I get into the mess I got into before, I'm not going to be able to build my credit score, and that I will probably get more offers since other companies will see that Chase extended me credit, and I should be careful.
He said I should get a welcome packet within 7 - 10 days, and asked if I had any questions for him. I thanked him for his help and explained how I found the number on the MyFICO website and that it seemed that they had more leeway than the automated system. He explained that it's not about leeway, but rather that they can get to know the potential customer and listen to their story. Most people are actually good candidates, they just messed up at one point or another. It's his job to figure out who is a good candidate and weed out those looking to dig themselves further into debt.
All in all the process took 12 minutes and 41 seconds to go from "No" to "Yes". On the first try even. Well worth it. Good luck to everyone else who gives it a shot!
