"Successful" is a relative measurement. To simply enjoy the process of learning and creating music can be done at any level and that to me is quite a successful relationship with music. One might not consider someone lingering on lower grades as being as successful as one who plays higher levels. That to me is almost like saying the poor can never be happy and the rich are always happy.
I've taught a 50+ year old who never played piano before and she was playing her favorite pieces amongst them Chopin concertos and Beethoven sonatas after a couple years. She also took up painting/drawing which she also never did before and created photo realistic works. A super talent who was uninhibited by age.
I have taught a few late 60-70+ who made modest progress in terms of their level of playing. Coordination was most of the times a limiting factor all of them however thoroughly enjoyed the level they were at and created good music, so they were having a successful relationship with music.
I've taught several self taught adults of varying ages and they all do quite well. It seems that those who actually try to teach themselves first for a while do better than those who come completely fresh with zero experience. They are much easier to improve because they have some foundation.
Those who have learned other instruments or learned music when they were younger also have advantages over those who didn't have any music education at all. Notably sense of beat and rhythm and coordinating your hands to stay in time can be very difficult for those who never had experience with it.
Those who miss out on childhood music really can have this lack of development which can be seriously inhibiting. It can be shocking but some people never did music even in their early schooling years, no music group games, never touched a musical instrument of any sort and never was sung lullabies as a young child's, raised in a desert void of music.
Topic: Teachers: If you have had successful adult students, what were they like?