Hello James,
Make sure to establish a screening criteria and apply it to each applicant without bias. If you are looking at an applicant's credit score, consider why they have bad credit. Look at their debt to income ratio, amount of 30/60/90 day late payments and types of delinquent accounts. Below is an example of a Risk Assessment that you may find useful:
Positive credit = Low Risk; accept applicant
Negative credit due to Educational Loan Delinquencies or Medical Payments = Med Risk; approve applicant with increased deposit/co-signer
Negative credit due to utility delinquency, money owed to a property management company or homeowner, car payment or unsecured debts = High Risk; turn down applicant
Unfortunately, FICO doesn't do a great job of considering the above Risk Assessment when scoring a consumer. You may want to try using VantageScore (made for tenant screening) or eliminating the FICO Score from your screening criteria altogether.
You may also want to consider refining your verification process. I understand the prior landlord references didn't have anything bad to say, but did they have anything positive to say? How did you verify the information on the application? Did you verify income with their employer?
Feel free to contact me directly with additional questions regarding tenant screening. I'm happy to offer tips. Good luck!