BOROMIR [to Frodo]: What chance do you think you have? They will find you. They will take the Ring and you will beg for death before the end. You fool! It is not yours save by unhappy chance... it might have been mine. It should be mine. Give it to me! Give me the ring.
At the climax of betrayal, Boromir seems to be making some pretty reasonable arguments. What chance does Frodo have? Not a good one. And why does Frodo get to carry it? Just because it randomly fell into the hands of his uncle? But Boromir, his will superseded by the Ring's will, falls into pure greed and grabs Frodo, trying to take it from him.
BOROMIR [to invisible Frodo]: I see your mind. You will take the Ring to Sauron. You will betray us! You go to your death and the death of us all.
Boromir is probably just saying outlandish things because he's upset at his failure to take the Ring from Frodo.
GANDALF: Tell me, friend,when did Saruman the Wise abandon reason for madness? [A brief scuffle ensues; Saruman is victorious.] SARUMAN: I gave you the chance of aiding me willingly. But you have elected the way of pain!
This is betrayal on a truly terrifying level. Saruman is the wisest of wise, the leader of the Istari. He is literally a god incarnate. Gandalf isn't just losing a counselor; Middle-earth is losing a great ally. Saruman's betrayal shows us that even the wisest can be corrupted.
GALADRIEL [narrating]: It began with the forging of the Great Rings. Three were given to the elves… Seven to the dwarf lords… And nine, nine rings were gifted to the race of men… For within these rings was bound the strength and will to govern each race. But they were all of them deceived. For another ring was made. In the land of Mordor, in the fires of Mount Doom, the dark Lord Sauron forged in secret a master Ring to control all others, and into this Ring he poured all his cruelty, his malice and his will to dominate all life. One Ring to rule them all.
This is by no means the first betrayal in Middle-earth, but it was one of the greatest; the betrayal that started two great wars of the Second and Third Age. You may be wondering why these elves and dwarves and men were dumb enough to accept some suspiciously nice presents from an evil overlord in menacing armor. But Sauron is a Maiar (a Middle-earth equivalent of an angel), after all, and can be especially cunning and deceptive when he isn't bashing heads in with a giant mace.
ELROND: Cast it into the fire. Destroy it! ISILDUR: No. ELROND: Isildur!
There's a reason Elrond doesn't trust the hearts of men, and his name is Isildur. Really, Isildur? "No"? That's it? How about some well thought out argumentation?
GALADRIEL [narrating]: And the Ring of Power has a will of its own. It betrayed Isildur to his death. […]Darkness crept back into the forests of the world. Rumor grew of a shadow in the East, whispers of a nameless fear. And the Ring of Power perceived. Its time had now come. It [the Ring] abandoned Gollum.
Hey, who says betrayal is only for living beings? The Ring itself gets in on a whole lot of betraying action when it leads Isildur to his death during the ambush, and when it abandons Gollum under the Misty Mountains. It really does have a mind of it's own.