Does this mean my Moller 71's are fake?  

The backs of Møller's 71 & 55 are made of bent plywood, with a layer of primary wood front and back, and a piece of edge banding on top. I have never seen this plywood delaminate, but it appears that perhaps yours is doing so? Perhaps from soaking in water or steamed somehow.
That is some serious and not easily fixed damage.

Oh that's worrying!! What should I do? Is there any way to fix them?

How loose are the pieces of wood?

They budge but you wouldn't be able to pull them off. The tops seem to be the worst affected, but stuck fast everywhere else. Any ideas?

Also they're a bit wobbly in the legs, in fairness they're a bit of a disaster......!

As Leif says the backrest is made from laminated timber with a veneer on each side and a thin strip of veneer on the very top. If you look underneath the backrest you should see the layered laminate exposed.
Maybe it is just my imagination but the very top strip of veneer on your chair looks unusually thick. I don't have a chair here to compare it to but I remember it being very thin and discreet on previous chairs I have had.

Thanks Howardmoon, in the chairs that are in better nick the veneer is seamless, you wouldn't be able to tell it was veneer at all. Gut feeling is they're genuine moller, they were heavily used in a boardroom for 20+ years and then used as dining chairs for another 20. I'd love to have a close up pic of someone else's Moller around the veneered part! A friend showed me theirs a little while ago but they're working abroad for 2 months. If anyone has any pictures if appreciate it!
Do people generally think mine are genuine?

Was the damage identified in any way by the seller before purchase? I assume they were bought online (or some similar manner where you could not inspect them firsthand before purchase).
If the price seemed too good to be true, and you bought sight unseen without asking appropriate questions in advance, then there may be shared culpability in this case.

Oh I take responsibility for buying them in this condition, as they were bought sight unseen! What would you consider agood deal for 10....? However if they are not genuine Moller then I feel I would have a come back as they were sold as such. I'd say I'm stuck with them tho, so right now I'm just trying to make sure they're genuine and then go about costing repairs etc.

I don't have any doubt that they are real. Like Howard Moon said, I also thought the top veneer was thinner, but upon looking at my own chairs, I see that some are the same thickness as yours, and others are a little thinner, depending on how much the finisher sanded that area at the factory.
If you got ten chairs, how many of the ten do not show this kind of problem? Sets of six and sets of eight are still worth a good amount. In any case, if the seller did not disclose this obvious problem, they were being intentionally deceptive in my view, as it is a significant problem that greatly effects their value.
I do not think these chairs have ever been "faked" so to speak. They are not easy or cheap to produce, and the profit would not be good for the company making the fakes. There are a lot of easier things out there to reproduce.

It's half and half, with the same for wobbly legs, unfortunately some overlap. I needed 8 so I'll need to work on some of them.....by that I mean get a professional I'd say! They were too good to be true.....but still enough to irk me!! I really appreciate all the help, talking about it is cathartic.....!

Ps tchp thanks for having s look at your chairs!!

I have 19 of these chairs. Well technically 11 now since I just sold off 8, but they are still here for a while longer yet. Yours are real. The top strip of primary wood is not veneer (at least not on vintage specimens). It is quite thick, maybe 1/8" or even 3/16". I can measure mine if you would like.
And if they don't have Møller stamps, they are the earlier ones, which are quite a bit nicer, in my opinion. And many, many of these are not marked.
As to the problem: you might see if you can steam off the veneer. I have a little steamer I use for disassembling things. It works quite well on PVA glue. Shoot he steam into the gap and gently see if you can pull it open wider and wider. Then you can clean the inside surfaces, remove excess glue (wet PVA does NOT stick to dried PVA), reglue and clamp with a LOT of clamps.
Most people would just work glue into the gap, then clamp it and call it good. I would avoid doing this because the glue will not hold well, nor for long. It is not a real repair. Not even a bandaid. It only makes the problem worse in fact by adding excess glue.
This is a pretty serious problem. Plywood is generally solid until it isn't, and then there isn't much you can do about it.
If this problem was not disclosed and you have any way to return them I would consider it. But not because they are fake. They are 100% real. You will never win an argument claiming they are fakes with anyone knowledgeable. Because they have serious damage.

As always Leif your knowledge and help is amazing! I'm afraid they are definitely mine now, the seller has disappeared off the face of the earth. As I didn't pay a premium for them it makes me feel a little better about it. I paid about how much some people pay for one or two chairs. I may just try and have them fixed one by one over the next while, try and restore them to their former glory? I'm hoping it will be possible, if not for all of them then at least for a set of 6. I'm at the stage where I can't really afford the nicer things I want unless I get them buy chance or there's something wrong with them!