You’re better off with a private landlord (in other words, someone who may have an apartment in a house or smaller building) rather than a large building or complex that will probably utilize the services of a managing agent. A managing agent is probably going to do things “by the book” meaning credit check, etc. They have a lot of experiences with deadbeats who might make a nice impression, and then try to screw them by not paying the last months’ rent and leaving a mess, stuff like that. If you get a smaller landlord, like someone who has a unit in their private house or someone who owns a small building and does the renting himself, he may be more sympathetic and casual about the arrangement.
I would say regardless of whether it’s a rental agent or a smaller landlord, the amount of money you have will influence whether or not they’ll rent to you. If you go to a rental agent who may run a credit check and you have, say, $10,000 in the bank, sure, you’re a good bet. However, if you have only say, $1,000 in the bank, they might not go for it. It all depends on the circumstances (i.e. who the landlord is) and how much money we’re talking. You did not specify how much money we’re talking.