The SFHT is a recognition that to make science fiction work it is perfectly acceptable to make up new science or to bend the laws of physics, but once you have set up a premise, it needs to work consistently and logically, or it fails the SFHT.
The original H. G. Wells invisible man (and many in the movies) have had a big problem because their alleged mechanism was a treatment that made our hero transparent. The SFHT says it's fine to invent a mechanism for making someone transparent - but then you have to live with the consequences. And for Wells' invisible man this should have meant going blind. If he had literally become transparent, light would pass straight through his eyes without interacting with them. So there would be no stimulation of the optic nerve - no vision. If, on the other hand, his eyes were allowed some kind of special treatment that prevented this, they would either appear as black holes in space (if they absorbed all light) or floating eyeballs (if they re-emitted photons).
|A cuttlefish working its chromatophores for all it's worth|
Now, SFHT says it's fine to allow this and to overlook all the difficulties of making it work to the extent of making a person invisible. (Apart from getting these alien structures in our skin, they would have to respond much faster than the original, be much more detailed, and be able to pick up a detailed image from the other side of the person, where an actual chromatophore user seems to primarily use information from its eyes.) However, once we have established this, the chromatophores have to act logically and consistently.
You could just about imagine a mechanism for concealing the eyes but leaving them working, though two spots on the back of the head behind the eyes would be visible unless there was a different mechanism for picking up the image from the pupil. But the real problem is with dead stuff. Hair and nails, for instance. Chromatophores have to be in a part of the animal that is alive. For nails this wouldn't be too much of an issue, because they are translucent, so they would produce ripples but be semi-invisible, but hair is a real issue. Doubly so, in fact. All your hair would be visible, and any part of your skin directly opposite a bit of skin shielded by hair would also be visible. It's a massive SFHT fail.
Interestingly the two real bits of invisibility technology echo these two science fiction approaches. The most effective at the moment is simply to put a lot of cameras on one side of the object you want to conceal and a screen on the other. Look at the screen and you see through the object. This is conceptually similar to the approach used in Fringe. At the moment it is an approach that is flawed because your invisibility only works when viewed from a single direction. But it would be surprising if, within a few years, we couldn't produce a spherical shield that consists of a matrix of alternating miniature cameras and LEDs, like those used in LED TV screens. The result is that you would both pick up and transmit an image in any direction - it should produce genuine cloaking.
The other, in some ways more impressive, technology, which you see quite often in the press, is cloaking using metamaterials. These are artificial materials that play around with the way substances interact with light or sound or electromagnetism. Invisibility metamaterials are usually those with a negative refractive index. You'll probably remember from school, refraction is the way light bends as it travels from one medium to another - causing effects like a bending pencil when one is put in a glass of water. Negative refractive index means that the light bends in the opposite way to usual, making it ideal to bend around something and conceal it. Like the original invisible man, with this kind of invisibility, the light never goes through an intermediate electrical signal.
The trouble with existing implementations is that they only work with very small objects and mostly with microwaves rather than visible light. The fact that this is reported in the media as 'Harry Potter style invisibility cloaks created in the lab' reflects the way the media (and university PR departments) can't help wildly exaggerating to get our attention. (A most dramatic example of this recently was the claim that a real Star Wars lightsaber had been made. Headlines literally claimed this. Actually what had been done is linking together two photons so they acted a bit like a molecule of light as they passed through a peculiar substance at near absolute zero. Not exactly a lightsaber.)
The kind of shielding provided by metamaterials, unlike the TV camera version (which doesn't render you blind as you can have inward facing LEDs as well) does suffer from the classic invisible man problem. As the light no longer passes through your pupils, but is deviated around your body, you can't see. Shame really.
So there we have it. You can be invisible and succeed with the SFHT. But very few actual examples in novels, TV and film actually do pass the test.
Image from Wikipedia