I was taught that it's best to learn small sections independently. That way you can dedicate an equal time to all of them, and play them all equally well. I certainly prefer playing or listening to a piece played moderately well than a song play partially great and partially horribly. And taking the song as a whole is just too much (unless it's very short). You make progress much more slowly that way, and are prone to getting into a rut where the bad playing has become a bad habit/muscle memory.
You do, of course, need to play the parts you've already learned while you're learning a new one, so that your playing of them doesn't begin to deteriorate. You should just play them enough to keep up that playing level, and focus on the new part.
I also practice playing the "joints", if you will. For example, if I'm learning 16 bars at a time I will play bars 9-24 after I've learnt both 1-16 and 17-32, just to ensure that my transition between them is smooth and natural. It shouldn't sound in the end like you learned them independently. And once you've mastered all the parts individually, it's important to practice the song in its entirety to ensure everything is cohesive.