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1.
When you become so comfortable with someone that you no longer even refer to them by their first name. I.e. being beyond first name basis.
Man: "Honey, I'm home!"
Woman: "Hey baby, how was your day?"
Man: "It was great sweetheart."
Woman: "What's my name?"
Man: "Ummmmmmmmm..... sweetheart?"
Woman: "What, are we on no name basis now?"
by Rothdaddy October 30, 2006
 
2.
Sharing time and/or space with someone whom you moderately appreciate or have absolutely no respect for to the point of not remembering or deliberately forgetting their name.

Pouet: So, who you out with last night?
Tuatil: Arh! A bunch of jerk-offs. I wasted a whole evening with a load of idiots.
Pouet: Really? Who?
Tuatil: Can't remember. I'm on a no name basis with'em.

or

Rebacci: How's your new roommate Jelad?
Jelad: Alright I guess. We're on a no name basis which shows how much time I spend at home with her!
by Berg Sloe September 10, 2007