Good Mistakes - It can be hard for teachers and parents to watch a student struggle through something without simply giving her the answers, but children always learn a skill better when they have figured it out them selves. Struggle is good. Mistakes that are born of conscious effort are great, because they are learning opportunities. Children often get frustrated by this kind of mistake and it's our job not to be frustrated with them, but rather to encourage them to keep working through it.
Consider Olympic gymnasts. What we see on our televisions are very complex and challenging routines that incorporate tasks we can't imagine being able to do ourselves (a back flip on a balance beam for instance), and those tasks are performed at a very high level. We know, however, that the gymnast on our screen has fallen off that balance beam, hundreds, or even thousands of times. With each fall, she has had to assess the problem and make small adjustments to where she shifts her weight, where she places her hands and feet, and so on. What we see in her Olympic routine is the culmination of thousands of hours of trial and error and discovery and repetition.
We need to allow music students to fall off the balance beam, and we need to encourage them to get back on and try again.
For insight into the role mistakes play in skill development, I recommend that you read Daniel Coyle's The Talent Code.
Bad Mistakes - The kind mistakes of which we should be wary are the mistakes born of carelessness. Often students will play something incorrectly because they didn't take the time to think and to prepare before starting to play. This usually means they are not focused on the task, and then it becomes our job to find ways to redirect their focus. Sometimes it means we have to change activities and come back to this task at a different time.
It's usually easy to distinguish a "good" mistake from a "bad" mistake, because when the mistake is bad, the student usually doesn't even notice it. Students are self-motivated to correct "good" mistakes, but if they are not paying attention - if they are playing on autopilot - they do not go back and correct things.
When we learn from a good mistake and correct it, the corrected version gets wired into our muscle memories, and then we strengthen that skill with each thoughtful repetition. When we make careless mistakes, the mistake itself gets wired into our muscle memories.