omg I have the EXACT same problem!! and I get it the same way- breaking when the index finger is too bent. I havent changed my ways because, well, Im stubborn, but I have found ways of living with it. First, I apply an ointment called "New Skin" to the place where I have the nail growing under the main nail, let it dry (and wait for the anesthetic to kick in), and if the nail is long enough, you can use pliers or tweezers to yank the obtrusive nail OUT --> completely!!! You may think this hurts and it will if you dont do it right (eg. the nail is the wrong length compared to the main one;) or if you pull at the wrong angle. You will have to pull from one side to dislodge it from the skin, and the rest is usually pretty easy.
Now, sometimes This technique has gone awry and it hurts like hell, so an easier half solution is to cut the obtrusive nail as short as possible and fill the gap with "New Skin". This way, you can play piano normally, but without applying New Skin it will kind of hurt. you apply the ointment everytime ytou practice until the finger heals.
If you have never heard of New Skin, it is kind of like rubber cement for your body. (hhmmm, I wonder if I could use rubber cement, hmm model airplane glue..i dunno)
donjuan