OK I see the problem now!!!
We're NOT talking about the same chord!!

You're talking about a chord with the 6th scale degree as an ADDED harmony, so that in C Major you ADD the 6th to the subdominant and omit the 5th scale degree? So, the subdominant root will then be F in C major, and the other notes of the chord are A and D?? Am I right?
This is NOT an N
6 chord or even related to it. Sorry to say, but the 6th you're talking about is completely different than the 6th we're talking about. The 6th we're talking about has nothing to do with adding any notes to any chords, but rather it ONLY describes what inversion any given chord is in.
A root position triad stacked in thirds (such as C-E-G) in C Major is designated as a I
53, the numbers to the side of the roman numeral describe the relationship of the bass note with the notes above it, in this case the interval between the bass note and the top note is a 5th and between the other note is a 3rd. This means the chord is in ROOT position. First inversion of that same chord is noted as I
63, or simply I
6, and the same logic applies for those numbers as well.
Enter bII
6, otherwise notated as N
6. The uppercase roman numeral means the triad is MAJOR, and the flat sign before it means the root is flatted. The
6 means, as in the last paragraph, that it is in first inversion. Therefore, in C Major, the flat 2nd scale degree is D-flat, and if you make a major triad becomes Db-F-Ab. First inversion is F-Ab-Db.
There are no added notes, and the 6th scale degree is nowhere to be found.
The root is the flat 2nd scale degree, and in first inversion you will never have the root as the bass note. It is impossible! The definition of first inversion is that the root is not the bass note, but rather the 3rd scale degree is the bass note, since the root has been moved from the bass to the top note, and is now a 6th interval above the new bass note.
I hope that wasn't too confusing.
Josh