If DC Navy Yard Shooter could get on a military base using someone else's ID, how could background checks have stopped him from buying a gun

From CBS in DC:

One gunman — identified as 34-year-old Aaron Alexis of Fort Worth, Texas — is also dead, and police are searching for a second possible attacker who may be disguised in a military-style uniform, authorities say. Alexis is believed to have gotten into the Navy Yard by using someone else’s identification card. It is not yet clear if that individual was an accomplice or if that person’s ID card was stolen. . . .