Yes, the big whites are allowed issues. It just seems unlikely that I had three copies of the 24 L II, all brand new, non of them worked, too exchanged under warranty by canon. I had two copies of the 70-200 mk2 and changed three AF units and Four IS units. One of them also had to realign a couple of the lenses! It never turned out okay., also under warrranty. Two 580 flashes with multiple issues. Had a 300 f2.8 that front focused about 100 meters aiming at 150 meters. All of this is very strange if it's only the occasional issue..
Jesus Joseph & Mary, that's an equipment run of bad luck I wouldn't want to own, really remarkable, I'm surprised Viggo that you haven't gone over to the dark side after this amount of issues. Hearing from others that they've had zero issues wont help, but unfortunately your 200f/2 is sick, until you get it checked you wont know why, if, as others have said, the Previous owner/yourself didn't drop it, take it for a sand bath, then it's likely it's a mechanical issue, which your warranty will cover if the Lens is less than 12 months old.
Cost of getting it into Canon & back unfortunately is a cost of ownership, something we are stuck with I'm afraid.
It sort of boiled me when after the cost of buying the Version II 600f/4/400f/2.8 & 300f/2.8, everyone of these Lenses had to go back to Canon Singapore for "Firmware" updates, due I believe to compatability issues with the 5DMKIII & 1Dx, at the time all the Lenses were at my Home in Jakarta, so it was a pack up & transport to Singapore situation, the "firmware" upgrade for my 200f/2, same situation (issue with the 5DMKIII), same for my 1Dx Bodies & the "Recall", it's just the risk factor involved in ownership of Equipment.
But I also owned the Nikon D800, and my copy had the left/right/upside down/whatever, focussing issue, Nikon "fixed" it 3 times before eventually replacing with a new Body, which I promptly sold, with no regrets.
I honestly hope you get the issue resolved satisfactorily, it's such a pity you've had these issues with what I would consider about the best Lens Canon have made, with the exception being the one you unfortunately own.