Being retired Law Enforcement...… a Police Records Check (local where you live) is simply a way for the employing agency (in this case the military) to see if you have pending or recent issues that will cause you a problem later on so they don't waste the time or money on a full Background Investigation. It gets you in the door so to speak, but DOES NOT grant any type of Security Clearance! The SF-86 or Request for Background Investigation is something a trained investigator (usually) conducts and includes NCIC, NLETS, and NADDIS Checks, PLUS the investigator will review your financial records, student loans, credit cards, spending habits. They may and usually do also speak with those you list on personal info forms as to what you were like as a young adult, who your friends were, and what your teachers and neighbors say about you. Then they produce a report to the requesting authority that details everything they found.
This process used to be pretty easy, but since all the big high profile security breaches in the last fifteen years things are much tighter. I think, after speaking with a person at USMA that the "police check" gets you in the door, but before you can be granted a Secret or above clearance the SF-86 or BI needs to come back clean. I have seen agencies require repayment of student loans or outstanding debt prior to accepting the appointment (both civilian and military) and granting clearances. There is a big issue with finances as the agencies are very wary of anyone with debt that may be susceptible to taking bribes or trading info for money. Same thing with any history of anything involving sex and money, or not being truthful on forms or during interviews, again the appearance of being susceptible to someone threatening to expose you. Our DS was advised that if you currently have an active Secret or above clearance from the federal government they will ask the agency that granted your clearance for the BI info if it is recent and use that to transfer your clearance.
Most applicants to academies should not have many issues, they usually don't have student loans, debt, or criminal history (hopefully), and the family and neighbor info is usually pretty clear and not an issue. it is the older people or people that moved a lot or had some employment issues that usually have a more difficult time. IMHO, I think if you are AD, have a clearance, and get a police check from your police at your duty station you will be good as gold.