From information I've garned here and on other Penfed threads, here are three things one should be sure of when applying for a Penfed card:
1) Your Equifax FICO NextGen 2 score - Penfed is pretty much the only major lender that uses this variant - needs to be at least 620.
2) You should be at least one year past the discharge date on your bankruptcy, whether Chapter 7 or Chapter 13 (Navy Federal wants this too). Also, other types of public records, such as garnishments, liens or civil judgments will adversely affect your application and may require you to contact Penfed directly for a reconsideration (see below) where you can explain in more detail what happened and why.
3) You should not have had too many hard pulls and/or accounts very recently opened on your profile. Penfed dislikes this, refers to it as "pyramiding debt", and will use it as a denial reason. To be safe, even if #1 and #2 are OK, if you've applied for/been approved for several other CC's lately, you should let those new accounts age and establish a payment history for at least a few months before applying to Penfed.
All that having been said, Penfed is notably generous when one contacts them directly via phone or in person to ask for a reconsideration on a denial. They appear to like being able to talk directly with an applicant to get more information straight from them. 