Most won't come out and say "there's nothing else I can teach you", so when do you know that you'd be better off on your own?
Generally this happens as your goals change. Maybe you started taking piano lessons just as a hobby (or because your parents forced you to) but are starting to take it more seriously now. Maybe you want to become a concert pianist, and your teacher doesn't have the knowledge or experience to take you there. When you start to feel like you're not getting anything from the lessons, you should find a new teacher.
Or is everyone better off with a teacher at all times?
A good teacher will not only teach you about the piano and about music, but will also show you how to teach yourself. So you shouldn't be dependent on your teacher.
It's always good to have a good teacher. But a bad teacher will hold you back. So know how to discern between them, and when to cut yourself free.
Also, are there any concert pianists still taking lessons while being as successful as they are?
I'm not sure about pianists. But I remember my violin teacher telling me something about the violinist Jascha Heifetz. Heifetz wouldn't play a piece publicly until his teacher, Leopold Auer, had heard it (and critiqued it) first.
- Saturn