Long before I ever tore into an automatic transmission, I used to hang around backyard mechanics (my dad and his friends), and although they new nothing about transmissions, they all new that any transmission problem could be solved with a band adjustment. In the "old days", automatic transmissions were very crudely built. Little was known about friction lining materials; Brakes, and transmission clutch and band lining wore very quickly, and needed frequent adjustment. If an engagement problem was felt, the bands were adjusted in the transmission, and very often, this cured the problem.
Now, today, the friction materials are much better, and they are much thinner. Transmission bands in today's transmission don't wear as much as they used to, so under normal conditions, they need to be adjusted much less frequently than the bands of older model vehicles, but it is still a good idea to check the adjustment during servicing. In addition, very few transmissions use a band for first gear engagement in "drive", so if a transmission will not move the car, chances are slim (actually, less than "slim") that a band adjustment/ service will fix the car. Band adjustment, however, can affect shift "overlap", (the amount of time, when the trans is shifting, that the trans is in two gears at one time) so band adjustment is still critical
For specifics on band adjustment, you will need to consult your own service manual, but generally, to adjust a band you will start by:
Because these new friction materials tend to wear very slowly, if at all, alot of manufactures have done away with the "band adjustment" during service intervals. I still pay close attention to the band clearance at overhaul time. (band clearance is adjusted, in many cases, with different length servo pins).