Here we go...
"Bach's appointment at Arnstadt was attributable to his reputation as one who had technical expertise in pipe organ design and construction. As a youngster in Eisenach, Ohrdruf and Lüneburg, it was Sebastian's good fortune that these towns had undertaken renovations of the organs in their various churches. In view of his instinctive curiosity and keen ear for the acoustical properties of instruments and interior spaces, there can be little doubt that Sebastian was a frequent companion of the guildsmen in charge of these reconstructions, if not an active participant himself."
From this page...
http://www2.nau.edu/~tas3/arnstadt.html#technicalThat passage makes a crucial distinction that I'd not thought of before...
Organ makers and repairers in those days were guild members. They had to be. It was highly skilled and in many ways was sort of a "closed shop" type of arrangement. Organs were, until the advent of the industrial revolution, the single most complex mechanical devices on the face of he earth.
There's no evidence at all that Bach ever apprenticed to a guildsman at any point in his life, and from the fair amount that's know of his life up through his 20s, when he began to receive patronage positions, there's simply no time in which he could have apprenticed for the term that would have been required to learn the art.