Why are we told we have a good chance with a credit rating of 600?

When your credit is sub-par (ours is about 600), yet the lender still wants to work with you, how much longer does it take than if your credit is good? We were told to put a bid on a house and that this would speed up the process. We bid on a $60,000 house (we live in a small town--homes are very cheap). My husband is the one with salvageable credit, around 600, due to a few credit cards from 2000. He needs to write letters to explain why they were unpaid. Everyone seems really optimistic. I'm really worried. Hub has a good, stable job and we have been spot on with our rent for the past six years and have no current debt. We can afford the mortgage if we get the house, according to them. I guess they are trying to get us a FEMA loan.
If we don't get approved, we are in a lot of trouble. Rent is higher than mortgages here...and we don't want to move into a trailer. We have two kids and don't think a trailer would be enough room. Thanks.