They may not be _totally_ independent, but the correlation coefficient is generally nowhere near 1. If you have reached the point where taste and skill are identical, then either you have none of either or you have reached perfection.

That being said, as your skill improves, your taste is likely to improve because you are better able to differentiate between high and low skill work. But in many cases, you don't need any skill to tell the difference. It's not exactly difficult to tell the difference between a mediocre amateur singer and a well trained professional. Where it gets really tricky though is that you may find that you may not prefer the most technically skilled practitioner. There are many people who would rather listen to Bob Dylan sing than Luciano Pavarotti. Is that poor taste? I'm not so sure. It's certainly possible to acknowledge Pavarotti's technical vocal superiority while still preferring Dylan. Whichever singer you prefer, you don't need any vocal skill to immediately tell the vast technical difference between the two. Many things are like this.