Having worked as an apartment manager, I can say that credit ratings were the single best factor for selecting a good tenant. I gave three tenants with bad credit a shot (single mom I felt bad for, blue color couple who seemed really nice and hardworking at screening and a couple who moved from abroad and had no credit at all) and I got burned by all of them ending up missing rent and needing to evict, sob stories and late rent, or causing other problems and fighting with their neighbors. After those experiences I basically only selected people above 750 to for our units and it always seemed like the higher the credit score yielded us responsible tenants who paid on time, didn't cause headaches at the building and left their units in good condition when they moved out.
As an aside, one interesting thing I noticed over the years fielding applications for open units, EVERY single person in the world who has bad credit or subpar credit apparently has the same psycho ex. It was rather amusing hearing that story over and over again. If you have poor credit and you're trying to rent a place- own up to it and be honest about why it's bad!