Yes, Wolverine can die. His healing factor has grown over the years to the point where he was able to survive being caught in an atomic blast and having all the soft tissue incinerated from his body, yet within a short amount of time he was back to normal. His healing factor also prevents his growing old, since his cells don't age, therefore don't die. The comics also have him as being the victim of a gang of black market organ harvests, who removed all his internal organs, by reaching from under his rib cage rather than through it, and he was able to regrow his heart, lungs, liver, etc, before he was able to die. It is unknown if he can drown. While it is believed that decapitation can kill him, this is nearly impossible due to his skeleton being made of adamantium—a fictional metallic substance that is orders of magnitude stronger than titanium and steel, and does not rust but can be magnetized. Also if he was decapitated his head would have to be removed from the vicinity of his body. Deadpool/Weapon XI, a character who was surgically given Wolverine's healing power, can and has survived decapitation, but needs to have his original head put back on his body for the neck to heal to two sections of him back together. Since his healing ability is based on Wolverine's, one can assume that Wolverine might have this ability as well. His healing ability also makes Wolverine immune to death from any type of disease, poison, or toxin, though a strong enough dose can make him sick until his healing kicks in.
Since Wolverines healing factor prevents the adamantium in his body from poisoning him, one way that could kill him is if he came into contact with someone like the character Leech. Leech's mutant power is that he can make other mutant's powers not work. It is reasonable to believe that if you used Leech to take away Wolverine's healing power, then he would be prone to death from anything that doesn't involve getting through his skeleton. In fact, this is supported by an event in the comics. At the end of the House of M storyline, the Scarlet Witch (who has the mutant ability to alter reality) utter, "No more mutants", and in a flash, many of the X-Men were transported back to the X-Mansion, only to discover they no longer possessed their mutant gifts. After realizing what happened, Cyclops said, "Oh no—Logan. Without his healing factor, the adamantium in his body—Kurt!" To which Nightcrawler replies, "On my way!", and transports to where Wolverine is, only to find him both alive and shockingly, for the first time since before the experiments of Weapon X, with his full memory intact. Other factors that have occurred throughout the comics can also turn off Wolverine's healing factor. According to the X-Men film, Rogue's powers could also kill Wolverine, by extracting life energy and his powers out of his body. Magneto could probably kill him by meddling with the adamantium in his body.
In the comic universe, there is a supernatural blade known as the Muramasa Blade which was forged and designed to be able to halt healing factors. In the comics, Wolverine uses this blade to decapitate and finish off his long time arch-foe Sabretooth, who had devolved into a such a savage animalistic state that he could barely speak and was even more of an unreasonable killing machine. It stands to reason that if a talented swordsman wielded the Muramasa Blade and inflicted a fatal blow upon Wolverine, his healing factor wouldn't be able to save him from death. In the comic series called Guardians of the Galaxy, which is set about 1,000 years into a possible future, the skeleton of Wolverine appears, but Wolverine himself is dead. In this future, Dr. Doom turned Wolverine's skeleton into a robot into which he transferred his own brain. There was a continuation of the Origins story called "Wolverine: The End" which takes place approximately 200 years in the future. Wolverine has aged to the point of being considered an old man (looking about 65 to 70 years old), his healing factor no longer works as good as it used to, and the adamantium has even begun to break down, most notably the center claw on his left hand having actually broken.
In the Marvel alternate reality series "What if...", Wolverine dies twice, once at the hands of The Hulk, and once on the surface of Earth's Moon due to lack of oxygen.