A "neck knife" is a small fixed blade knife that is worn on a necklace and concealed under a shirt. There are a lot of quality custom knife makers and knife manufacturers producing good quality knives for this purpose, and there are a lot of people buying them but I am not sure why.

Neck knives are so well concealed that the user can't even find them. Generally they are worn around the neck, blade up/handle down within a sheath under a shirt. This makes for a fairly complicated draw, that isn't too bad if you can use two hands: one to move the clothing and one to find the knife. A two handed draw is going to be difficult to pull off in a ground-grappling situation, or when the assailant is already at contact distance.

The necklace isn't helping. One school of thought thinks you should have a break-away type necklace so you don't get choked by it. The problem with this is that if it does break away you may no longer have the knife, or have the knife in hand with the sheath still on it. The other school of thought uses something strong like parachute cord for the necklace. I guess I am in the parachute cord camp, because if I am getting choked, at least I have a knife to solve the problem with.

Your knife isn't where you think it is. Because the knife is basically a big pendant hanging from the necklace, if you accelerate and stop the sheath and knife are going to continue moving. This means that if you fall, the knife is probably going to be tucked under an armpit. The longer the necklace the more movement the sheathed knife will have. If you are falling in a fight there is a good chance that the assailant is going to end up on top of you. This is going to make that draw even more complicated.

This is not your grandfather's pocket knife. 99% of the time that you are going to use your knife it isn't going to be for self-defense. While the knife is certainly capable of doing the job, the method of carry and draw stroke are unorthodox enough to raise eyebrows in every social situation. This is a large enough liability that you will probably need to carry a totally different pocket knife for general purpose uses.

That thing isn't legal. In most jurisdictions carrying concealed fixed blade knives is going to be against the law. While some will argue that the legalities are of a lesser concern than personal safety, I would point out that you would be completely legal if you were simply carrying a folding pocket knife. The folding pocket knife is just as functional, can be deployed in roughly the same amount of time from a variety of positions, can be used for utility without negative perceptions, and doesn't have any negative legal connotations.