It would all depend on the requirements of the Management Company. Some allow no collections what so ever, paid or not, some say no money owed to any rental property. Some say your credit score has to be 650 or above.
All of them are different and can require different things to qualify.
My credit is not good at all. What I have done in the past and will do in the future is be totally honest with them, before I fill out the application. I find a place I am interested in and go talk to the owner/manager/management company. I leave nothing out, tell them my scores, what is on my credit report, let them know I did not just not pay my bills, that sometimes you have to rob Peter to pay Paul. I have found that being completely honest upfront is going to go farther than letting them see a credit report they may not like.
I also toot my own horn, so to speak. My credit may suck but I am a decent person with a more than decent job who can afford to pay what they want. I let them know I will take care of the place as if it was my own.
Also, the probabilty that more than one company use the same screening company is high. If you really want the place you were turned down for, go in person to talk to them, if possible. If not, send them a letter.
