Like other contributors I think it all comes down to how you read the following passage, which appears shortly after the Attack on Frodo & party at Weathertop.
(It's the passage others have quoted, and I think I can explain specifically why there can be multiple interpretations)
"'Look!' He cried; and stooping he lifted from the ground a black cloak that had lain there hidden in the darkness. A foot above the lower hem there was a slash. 'This was the stroke of Frodo's sword,' he said. 'the only hurt that it did to his enemy, I fear; for it is unharmed, but all blades perish that pierce that dreadful King..."
Reading this passage one way, you understand Strider to mean that Frodo must have struck the cloak but missed the Black Rider. In this reading, Strider's is arguing that if Frodo had struck the rider his (Frodo's) sword would have been damaged - and so by inference, Frodo's sword is fine. I think now that this is the correct reading.
Formerly, I read this passage differently, as if Strider was meaning "look; you struck your enemy. But you won't have hurt him [i happen to know from lore I'm not going to quotes references for right now]. All you will have achieved is to ruin your sword." This is the meaning you get if you think that 'it' in 'for it is unharmed,' refers to the Black Rider, not the sword.
Having (mis)understood that Frodo's barrow sword 'perished' in his fight, I was then wondering what sword he had to break at the Ford a bit later. Also why Frodo doesn't get Ill the way Merry does later when he (Merry) does wound the Nazgul. Presumably Frodo's barrow blade would have been capable of wounding a Black Rider, just as Merry's does later - Frodo simply missed.
By the way, I have sometimes wondered why the Balck Riders don't press their attack on Weathertop - it proves to be their best chance to get the ring before Frodo reaches Rivendell. I found:
Tolkien quote - They felt they'd had a narrow escape from one of te few weapons that could harm them
http://newboards.theonering.net/foru...=274144#274144
A good discussion of this chapter, especially ideas about the limitations of the Black Riders' powers
http://newboards.theonering.net/foru...guest=20181802