Why was Boromir allowed to join the Fellowship of the Ring after the Council of Elrond, where he succumbed to the Ring moments after first seeing it, and explicitly said that he wanted to use it himself? It was made absolutely clear that he could not be trusted around the Ring, but no one spoke up to say "Maybe the guy who succumbed to the Ring within seconds shouldn't go on the mission to destroy said Ring".
Assuming that it was deemed necessary for the Fellowship to include a representative of Gondor (although Rohan didn't get a place in the Fellowship, for some reason), couldn't Aragorn fill that spot? If not, wouldn't it have been worth the delay to send Boromir back to Gondor and request that Faramir (who later proved himself to be less susceptible to the Ring) take Boromir's place? Granted, the Council had no way of knowing ahead of time that Faramir was stronger than Boromir, but it seems like Boromir was the worst possible choice, and anyone who wasn't Boromir would have been a better candidate.
Updated May 24: I am finally reading The Return of the King, and the heated exchange between Denethor and Gandalf just before the siege of Minas Tirith seems to support my argument. Denethor is angry at Gandalf for letting the Ring be sent to Mordor with Frodo, and says that it should have been carried to Gondor by Boromir instead, so it could be hidden. Gandalf says that if Boromir had taken it, he would have been corrupted before he ever reached Gondor and Denethor would no longer recognize him. Denethor accuses Gandalf of having stolen Faramir's loyalty and claims that the real reason Gandalf didn't trust Boromir was that Boromir followed Denethor's orders rather than Gandalf's; he says Gandalf doesn't trust Denethor's judgment..
"... I do not trust you... And now hearing you speak I trust you less, no more than [I trusted] Boromir."
- The Lord of the Rings; The Return of the King, Book V, Chapter 4: "The Siege of Gondor"
There we have it- Gandalf never trusted Boromir, or at least didn't trust him by the time the Fellowship was being assembled. Why, pray tell, was he allowed to go anywhere near the Ring, when the wisest person around didn't trust him to begin with?