The claim is a producer under pressure to meet time constraints made the edit. It's true, network TV news, heck all broadcast news, is a slave to the clock. But law #1 in editing, video or audio, is you NEVER edit when it will change context, which this clearly did. Regardless of what pressures the producer was under, surely he/she knew this would change context in a very inflammatory way.
Is this explanation possible? Yes. Is it the most likely explanation? My personal opinion is absolutely not.
UPDATE: This link shared by a listener would seem to indicate that the boat carrying NBC's story that an producer under deadline made a bad call has a lot of leaks in it. NBC had been using the truncated version, multiple times in online product? Before the broadcast? Clearly this had become the accepted NBC version of the conversation before they used the audio that way.