I purchased my black Note 3 the week it was released. The Home button tolerance was loose on the white store demo phone and on my new Note 3 - no difference between the two by my observations. Though I am somewhat OCD I chalked the loose tolerance up to design intention (I've not read anything definitive on the subject since - this is merely my assumption). There were no visible physical flaws beyond the noted lateral play in the Home button.
I noticed issues with the proximity sensor within few days. T-Mobile sent a replacement overnight and the replacement phone exhibits the exact same Home button loose tolerance as did the store demo and my first Note 3. The replacement showed no issues with the proximity sensor and no visible physical flaws.
More than two months after I received the replacement phone from T-Mobile everything works as it should, no chrome has chipped away from the sides or edges where plating is present, no dust has appeared under the glass covering the sensors or lenses, and the Home button has not changed in function or tolerance. I put this phone through heavy use daily with no case (save for a standard leather holster) and no screen protector. Assuming the proximity sensor continues to function properly for the remainder of my time using this phone I'll call this one a win.
All that said I consider myself a sample set of one - in other words: insubstantial on its own. I'm referring more to the faulty proximity sensor in the phone I purchased at the T-Mobile store - not the Home button. The loose home button may well be an unintended loose tolerance in design. Then again it may have been designed this way purposefully (my OCD has a tough time with the idea of intentional by design - such is the nature of OCD). Either way the button so far seems to function just as it should. I am sure there have been a few button function failures but so far this would appear to be a statistically insignificant occurrence even among the small sample set of those reporting loose Note 3 Home buttons on forums (compared to the total number of Note 3 phones sold to date).
I won't presume to discount the concerns expressed here and on other forums regarding the loose Home button. To my knowledge Samsung hasn't officially addressed these concerns with any admittance or explanation so these concerns may ultimately prove to have merit. ...or not - only time will tell.
I will admit that I was initially put off by the considerable effort needed to remove the S-Pen from the phone. A comparison between my Note 2 and my then new Note 3 made the Note 3's S-Pen seem ridiculously difficult to remove. As I had with my Note 2 I have since made heavy use of the Note 3's S-Pen and it soon settled into what I would now call an appropriately snug fit (presenting no difficulty in removal / insertion), a fit I now prefer over the Note 2's comparatively loose S-Pen fit.
No phone will ever be perfect. I think it's fair to say that the Note 2 received less QC or failure criticism than the Note 3 has to date on the various Android phone forums (this has been the case at least by my observation in rather heavy reading on many such forums since we'll before the Note 2's release). Even if we are observing a statistically tiny sample set it does seem to indicate some initial QC issues not observed of the Note 2 by the same group. Given the reported issues with dust between sensor's / optics and glass, faulty proximity sensor's, and some odd cases where the screen glass has cracked (potentially attributed to charging and/or processor heat). I take no issue with those expressing these concerns, nor do I take issue with those upset with any perceived flaw in any electronic device in the Note 3's price range. There's room here for everyone's perspective and experience here. This is a user forum. It would serve little more purpose than a generic chat room otherwise.
Hopefully Samsung gets any existing QC issues resolved quickly. From my experience thus far the Note 3 has amazing potential.
There's my $0.02 on this subject.
