The worst mistake a music teacher can make:
Deciding that a student "just doesn't have it."
"Alas, this poor soul was born without the talent necessary to become a musician. I'll collect my fee, but I really can't do anything with this one."
This is such a depressing idea. If this is where you begin, why even bother being a teacher? And if this is where you end up, you are seriously burned out and should take some time off.
There have been times in my career that I've struggled in finding the best way to reach a student. There have been times when I was frustrated at a student's lack of practice or progress. But instinctively, for self-preservation as much as compassion, I've avoided thoughts that allow me to take myself off the hook and blame the student.
Once you go down this road, it's very difficult to find meaning in the work of teaching. All of a sudden, you're deciding the destiny of a student based on your assessment of ability. You become a judge, rather than a teacher.
This is an incredibly destructive attitude. For many teachers, "None of my students has the dedication and talent I had," becomes a self-satisfied, self-fulfilling prophecy that stunts their own growth as educators and musicians. Obviously, the student herself suffers as well, since she's put her trust in a mentor who doesn't believe in her.
If you are a teacher who is tempted into this thought process, it might already be too late for your student. However, you can work on it. Consider that the student might have a unique learning style that you are not tapping into - for example, he can learn very well by ear and has trouble reading notes. Or he might have certain personal qualities, such as determination or a great attitude, that will make up for a supposed lack of talent.
While it is unfair to blame failure on the teacher and credit accomplishments to the student, neither is it acceptable to blame a student for lack of aptitude and at the same time claiming faultless teaching methods. Focus on yourself and not the student, and do what you can to improve your own work.
Whatever you observe as weaknesses on the part of the student, figure out how to strengthen their underdeveloped ability by breaking its necessary elements into achievable steps. Many things that people dismiss as innate talent (sense of pitch, sense of rhythm, expression, "feel") can be explicitly, systematically taught by a teacher who is able to calibrate her expectations down to very tiny increments of forward progress.
In acknowledging that anyone can get better at music, you might have to confront some uncomfortable realities about your own talent, accomplishments, and weaknesses. You don't have to be stuck where you are, either.
Somewhere along the line, you might have encountered a judge masquerading as a teacher. It's time to silence that critic, and in so doing, let a more compassionate, creative voice speak up. Instead of, "maybe she just doesn't have it," how about, "I charge myself with the responsibility of awakening this person's musicianship."
Harder, and worth it. If you can't do that, maybe you just don't have what it takes to be a teacher.