Unfortunately has proven after delivery - and a subsequent research on the Internet has confirmed - that Tatonka has apparently recently minimally reduces the internal dimensions of the cup, so that the cup can no longer attach from the bottom to a Nalgene bottle. Very bad! I'm interested in known earlier.
As "Replacement Emergency configuration" I can place it on the bottle the cup at least one centimeter overlap from above, so on the bottle neck, to about the top of the measuring scale. Is not so sure and not so compact and dirty cups maybe not quite as appetizing. I'll keep the mug but only once.
ADDENDUM: GSI Outdoors Glacier Stainless Steel Bottle Cup - later purchased ... fits; as usual, at the bottom of the canteen with seating for two, three layers of tape around the bottle to fill the tiny gap between the cup and bottle. So the cup can not easily slip off. Thus the fault does not lie on the Nalgene bottle.
The Tatonka Mug comes in better quality ... better and thicker stainless steel, better grip (part handles cross easily and are easier to handle) and with pre-made level mark inside. But if he just not as good fits ...