While it could mean both that he had failed to save his people and/or Merry and Pippin, I had always thought it could mean that he had failed the test of the temptation of the Ring. Gandalf did mention that Boromir repented before the end and that could easily be referring to:
Quote:
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I tried to take the Ring from Frodo. I am sorry. I have paid.
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... but I also feel that his admitting his failure was a part of his repentance. What I find more interesting is the fact that Aragorn admits
he fails right after Boromir repents. How much did Aragorn secretly depend on Boromir is something I've always wondered.