Am I the only person who has no idea what they're talking about? I've *never* had the music cut out on me. O_o
What they're referring to is the fact that the game won't play music constantly if you stay in a location for a long period of time. For example, when you exit the farmhouse in the morning onto your farm, a song will begin playing. If you stay on your farm long enough for the song to play all the way through, at the end of the song you'll just have the background sounds of "nature" in the game. The music can change when you move to a new map, but doesn't always. It's possible that some of your confusion is from their use of the term "cut out", when it would be a bit more accurate to say that when a song ends, another one doesn't start.
It definitely seems that some people notice this much more than others, I do notice when there isn't music playing in the game, but it's never bothered me. (Minecraft, another one of my favorite games, has a similar approach to music).
I will say for those that are bothered by this, you can at least use the
Mini-Jukebox to get looping music on your farm or inside buildings on your farm.