I'm not an expert, but, here is my understanding...
One of the things that is done during N-400 processing is that the USCIS passes your info to the FBI and says "Here's John Smith, can you check to make sure he's not a criminal or a terrorist or..."
If you have an unambiguous name, and you don't have a namesake who is infamous (John Dillinger, Osama bin Laden), the FBI comes back about 72 hours and says "John's ok". However, if you have a very common foreign name, or a name that has many transliterated variations, or there is someone with the same name as you who has done bad stuff, you fall off the fast track.
Instead of 72 hours, suddenly the name check takes weeks, or months, or years.
All this happens in parallel with the normal USCIS processing (most of which consists of letting your file collect dust, but...).
Some district offices don't interview if the name check process is not complete. One of the reasons for this is that some law (section 1447b of something) allows you to petition the courts if your application gets hung up too long after the interview.
Other District Offices continue their processing even though the name check process has not finished. Once the interview, the namecheck, the fingerprint check, and a few other things are finished, they make a decision and either issue an oath letter or deny your application.
HTH
