It isn't just Logan's adamantium-coated skeleton that's kept him kicking for so long. His other mutant power is his healing factor, which is definitely getting weaker in his older age, as the trailer for Logan suggests. Director James Mangold confirmed that the character will be in a new, more fragile state than fans are used to when we see him next, telling Empire Magazine that in order to establish a "different tone of a Wolverine movie," they had to make some key changes to the character, including limiting his invincibility.
"We felt like we couldn't hold on to every tradition established in all the movies religiously, or we'd be trapped by the decisions made before us," Mangold explained. "So we questioned whether Logan's healing factor causes him to heal without even a scar. We imagined that it may have when he was younger, but with age, he's getting older and ailing. Perhaps his healing factor no longer produces baby-soft skin. So we imagined he heals quickly, still, but it leaves a scar. The simple idea was that his body would start to get a little more ravaged with a kind of tattooing of past battles, lacerations that remain of previous conflicts."
Indeed the wear and tear of time has taken its toll on his back in the preview, and while that's not a death knell in and of itself, it certainly indicates that Logan is coming into this thing more vulnerable than ever. If his skin can bear damage, then, in theory, the rest of his body can too, right?