It depends on how the home owner/management company is doing their check. Many these days use an automated system that spits out a pass/pass with conditions/fail, and each of those systems behave differently. I do a lot of residential and apartment leasing in the area and have a pretty good read on what each of those systems are going respond with based on the situation.
It also depends on the area quite a bit, you can go out to a place like Manor and rent as long as you have a heartbeat and some income, closer to campus or near Zilker/Barton Springs they are much stricter on requirements. If you shoot me an email with exactly what you are looking for (area, price range, etc) we can get you setup with homes that would best fit your needs.