When Frodo was fleeing the Nine Black Riders and had crossed the Ford of Bruinen onto the Rivendell bank - with The Nine still on the opposite bank - there is a scene when Frodo draws his sword and shouts defiance (rather feebly). But the Witch King merely gloats, raises his hand, casts a spell, and the sword breaks in Frodo's hand.
This broken sword is one of four apparently identical weapons that Tom Bombadil gave the hobbits from the treasure horde accumulated by the Barrow Wight; after Tom had broken-open the barrow. They are described as long, 'leaf shaped' daggers for use by Men - but the right size to be swords for the hobbits. These were Numenorean blades, from the Kingdom of Arnor.
Later, on the eve of The Fellowship setting out on their quest to destroy the One Ring; Bilbo apologises that he forgot to have Frodo's broken sword mended; and gives him Sting instead - an Elven blade originally from Gondolin. This is a symbolic gesture of handing-on responsibility; and has important repercussions later - for example this type of Gondolin magical blade was made for use against giant spiders (as we learn in The Hobbit), and Sting was able to cut through Shelob's massively-thick web - when Sam's Numenorean blade could not.
But why was the Witch King able to break Frodo's Numenorean blade when, as we discover later, these were magical blades specifically forged to destroy the Nazgul Chief himself? (When he was known as the Witch King of Angmar, and was attacking the northern Numenorean Kingdom of Arnor.) How could the Nazgul Chief destroy by magic, a magical anti-Nazgul weapon?
The problem arises because it was Merry's use of exactly such a blade in the Battle of Pelennor Fields, that enabled the Chief of the Nazgul to be killed - it is described as undoing the spell by which the wraith's body was kept-together; such that a follow-up thrust from the ordinary (non-magical) blade of Eowyn was able to 'finish him off'.
The reason, I suggest, was because Frodo had been stabbed by a Morgul Knife on Weathertop - and this was a black magical weapon that had the effect of making Frodo a wraith. This wraithing process was far advanced by the time that Frodo crossed the Ford of Bruinen - Frodo was even becoming transparent, and falling under Sauron's control.
I assume that the Morgul Knife was indeed magically forged by Sauron - a very rare - possibly unique at the time, difficult to make, sentient weapon (although one that Glorfindel recognised, so presumably not the first such weapon) - designed so that the tip would snap-off after stabbing then - under its own power and sense of direction - work its way to the heart. Once the knife tip reached the heart, the wraithing process would have been complete.
The intent of the Witch King, according to Gandalf, was indeed to stab Frodo directly in the heart; making him immediately a wraith. The plan of the Nazgul was, apparently therefore, to capture Frodo and take him to Sauron as a new 'ringwraith' - and unable to die; where Frodo would have been tormented 'forever'.
This explains why the Witch King did not try to kill Frodo (e.g. with a sword) when he had the chance. Sauron did not want Frodo to die, but to be kept alive for the worst and longest-lasting punishment Sauron could devise - for having taken the One Ring and kept it.
But this wraithing had not quite happened by the time the Ford to Rivendell was reached; because Frodo had, at the last moment, resisted the Witch King's attack, and spoken the name of Elbereth - momentarily dauting and distracting the Witch king, so that the stabbing blow missed the heart by several inches - enough to delay the wraithing (in a Hobbit, which is more resistant to the corruption of evil than a Man) for several weeks.
Nonetheless, Frodo was most of the way towards being a wraith by the time of the Ford of Bruinen, and he therefore had a 'spiritual connection' to the ringwraiths - a channel had been opened-up. Frodo could see and hear the Nazgul much more clearly than could the other Hobbits, and indeed by this time he saw the Nazgul more clearly than he saw his companions.
Presumably, therefore, the Witch King was able, via this magical connection, to destroy even Frodo's magically-anti-Nazgul sword, by an unspoken command and a gesture.
This breaking of the Numenorean sword then created the desired plot-opening by which Frodo needed a sword at the last minute, and provided a reason for Bilbo giving him Sting.
Acknowledgement: To my son Billy, for the suggestions and conversation which led-up to this insight - putting to rest a long-term uncertainty.