So my SO applied and was instantly denied the Citi Costco card. They then sent a mail letter stating the credit score they pulled was 683 (from experian), and that he had limited history and a serious delinquency.

I think the limited history has to do with the fact that--despite having a credit card for 5 years, he has spent it maybe once or twice. So the "history" is there, but there is no data to report.

As for the "serious delinquency", this is in regards to some missed payments (3 if I count correctly on the experian report) on his student loan almost 7 years ago (2012).

I heard that derogatory remark may disappear after 7 years (?), so if we wait a few more months, maybe he would have a higher chance?
I also encouraged to start using his credit card (just with a local bank) to get some more history.

We are thinking to try applying for him again in 2 months. Is this too soon?
If he gets denied again, how serious/bad does it affect his credit or potential to apply for other cards?