I received an honorable discharge in 1991, due to a locally imposed Bar To Reenlistment. I got the Bar for getting in too much trouble in a short time frame (alcohol abuse, fighting, telling a CPL "Fuck 'em," about a SGT). My RE code was RE4.
After I had been out awhile, I looked into reenlisting. I was told that it was possible, but would require a waiver, signed by the recruiting company commander. I had to get several letters of recommendation from respected members of the community. For those, I chose a high school teacher, a manager at work, and the Adjutant of the American Legion Post where I was a member, who happened to be a deputy sheriff and retired Navy CPO. I then had to interview with the commander, himself. He asked about my alcohol use, if I'd been in any legal trouble since my discharge, etc. He then stressed to me that if he put his name on a waiver, any thing I did while in the Army would reflect on him and his judgement.
It took me five years, but I was able to overcome an RE4 and reenlist. But, that was in the mid-90's. The times are quite a bit different now. Then, no one wanted to join the Army and the recruiters were all but having to beg on the street corners. These days, the military is at the beginning of a major drawdown, and recruiters are turning people away for the smallest reason.