The National Institutes of Health has simplified the NIH policy for late application submission. Effective for applications submitted for due dates on or after January 25, 2015, there is a two-week window of consideration after the application due date, during which time NIH might consider accepting a late application.
NIH provides more details in the December 17 NIH Guide Notice, Simplifying the NIH Policy for Late Application Submission (NOT-OD-15-039). The guidance includes examples of reasons why late applications might be or will not be accepted.