Why doesn't the music in Minecraft play all the time?

It is normal for the music to fade out, but it should come back on later; a random music track is played at sunrise, sunset, noon and midnight. There is no easy way to have the music play all of the time.

C418, the composer who created the music for Minecraft, talked about this (and the reason for it) in an interview:

You also had the limitations of the sound engine to deal with.

Yeah. A couple years ago if you had two song files at once, it would actually crash. There were so many more weird glitches like that the guys never really fixed because they were too busy with the actual game and not the sound engine.

Did you try to take advantage of those limitations?

Yeah. One of the things that we really can’t figure out is what a player is doing. Is the player in a cave? Or is he in a house that he made himself? So I decided to basically have music that doesn’t really explain anything. It doesn’t say if it’s battle or if it’s night... Generic music, but still kind of unique and different. Then we had that music randomly playing 15 to 20 minutes in, where there’s nothing else playing. If something significant is happening, the player remembers the music associated with it. If the music is playing and nothing really is happening, they’ll just accept it. Originally I was like, “You should probably play the music as few times as possible because it’s probably going to be annoying.” And it turns out the pause in between actually helps.

The main point of the occasional nature of music in Minecraft is to build atmosphere. Minecraft is a, somewhat, lonely and isolated game (if you're not playing with tons of people yelling over Skype) so the game tends to be quite silent and atmospheric. There is no way to make the music play all the time since the sound design of the game is created in such a way that music plays at certain events, whether it's combat, nightfall or a sunrise. To look at it a different way, Minecraft is supposed to be, actually, a really creepy game (and if you are playing alone, it certainly is). So that answers the question of why Minecraft music is not playing all the time (of course, including colorfusion's answer about the technical limitations). As far as making it play all the time - no, you can't. I've searched for mods that do that to no avail, unfortunately.