It didn't happen while playing back in Final Cut; only when I exported the clip using the Using QuickTime Conversion option..
In the past I've found that odd things like this can sometimes be "fixed" by adding a "non-filter" to a clip i.e. forcing the editor to do some processing to the clip before it exports it. I tried moving the clips to another track, as well as putting another clip underneath the clip. All I seemed to do was cause the odd rendering effect to move to another part of the sequence!
I even tried copying and pasting to another Sequence, and exporting to Compressor (which takes about four times as long) and still the problem occurred.
Finally, I did a simple export of the entire movie as a QuickTime Movie, which saves it in the format of the Sequence, rather than recompressing to a format of your choosing. This finally worked; and then I just imported the clip into another Sequence and exported it again with the settings I wanted.
It was troubling that I couldn't figure out why it was mis-behaving; especially as the project was short and not very complex.
notesonvideo: Shooting a Music Video: Air Traffic Controller