I've had the same problem with my bamboo-backed Moto X 2nd Generation Pure Edition. I purchased the phone brand-new from Motorola's website late last September, and I noticed the defect about 2 months ago (so 2 months of warranty coverage left at that point).
Motorola support confirmed that the repair would be covered after I sent in an image.
They offer an Advance Replacement option for a $25 fee (non-refundable); when this option is selected, they'll send you a new phone while you keep your current phone, and then you send your old phone back after you transfer all your settings/data. The free repair/replacement option requires one to send in their phone and wait for the replacement.
Unfortunately, I've had a terrible experience with the Motorola repair department. I don't think I'll ever buy another product from Motorola, smartphone or otherwise. Over the past 6 weeks, I've received 4 (yes, FOUR) replacement Moto X devices from Motorola, and each one has had the same defect: a peeling bamboo back cover. I tried pushing the back cover down, but it always pops back up after several minutes (especially when the phone warms up a bit after several app installations). I spoke with a repair escalation department supervisor (Ruben) today, and he said he'll escalate this and assured me that the next replacement I receive (the fifth replacement) will not have any problems. I'll post an update upon receipt.
In case the fifth device is, yet again, defective, does anyone have recommendations for what I should do next (other than trying to fix it myself)? Does anyone have information on how I can send an email to Motorola's president, Rick Osterloh, or to someone who heads up the the repair department/oversees the supervisors?
In the meantime, I recommend not buying a Moto X (any generation). Some percentage of mass-produced products will always fail, and Motorola has shown that they don't respond adequately, even when given multiple opportunities. We'll see how the fifth replacement goes...