Okay, so you just brought home your "new" HammondOrgan. You turn it on, but get no satisfaction. The tubes light up, so you know it's got power, but you can't get the ToneGenerator to spin.
(First off, are you operating the Start and Run switches correctly? See HowToStartAHammond.)
When the start motor gets energized its shaft pops out (or at least it's supposed to) so that the gear on the start motor engages a gear on the main tone-generator drive-shaft.
The gear on the main shaft of the tone generator which the gear on the start motor engages slides on its shaft. It is spring-loaded so that normally it is slid "out" (towards the run motor). The idea is that when the start motor is first turned on and its gear pops out, the mating gear can slide out of the way momentarily until the teeth line up and the gears engage properly. (It is actually even more involved than that. When the TG gear is slid in, it can spin freely, or "freewheel", on its shaft. In the normal, "slid out", position, the gear engages a pin which keeps it from freewheeling, thus enabling it to drive the tone generator.)
The alignment of the start motor is critical --- if it gets out of whack the gear on the start motor can have trouble engaging with its mate on the main TG shaft. (One way the alignment can get screwed up is if the end plate of the TG, onto which the start motor bolts, get bent.)
Solutions
Another common cause of problems is that the gear on the tone-generator shaft can get gummed up so that it doesn't slide on its shaft freely. (Often it will get stuck in the "pushed in" position, making it impossible for the start motor gear to properly engage.)
Usually it can be unstuck by applying some HammondOil directly to the problem area, and working the gear back and forth on its shaft. In more stubborn cases, feel free to use some WD-40 to unstick things --- just remember to follow up with some HammondOil after it is all unstuck (otherwise the WD-40 will soon evaporate, and you'll be stuck again.)
Access
You can _sort of_ get at the start motor and the associated gears from the back of the organ without removing any parts, but it tight. Often it helps to raise the manuals --- that gives you a bit more room and makes it somewhat easier to see what you're doing.
Links
George Benton has a write-up (with great pictures) on the start motor system at: 
http://bentonelectronics.com/hammond-generator-start-motor/
If the start motor seems to be engaging correctly, but the tone generator isn't spinning (or doesn't get up to full speed), you more than like have a gummed up tone generator.
(Try to rotate the tone-generator by hand.)
Sometimes all that's needed is oil and patience. Oil the tone generator and wait (see HammondOil.) It can take days or weeks for the new oil to crawl it's ways up all the little capillary threads to the appropriate bearings.
Michael Fulk says that Hammond recommended the use of WD-40 for lossening or temporarily lubricating the TWG.[also an instant way to get rid of "bearing sqeal"
Some people have reported that WD-40, applied directly to the problematic bearings can loosen a stuck generator. Make sure, after you get it unstuck, to properly oil the TG with HammondOil.
Michael Fulk says the preceding paragraph is most accurate, pursuant to information obtained from Alan C. Young.
This may sound strange to some of you, but I own a 1936 Hammond model A which sat
unused for 30 years. when I bought it the tone generator would not spin up to speed.
The run motor was the problem. I oiled the organ for two weeks and still could not get
rid of the terrible bearing squeeling. A Reverend from the mid-west told me to make a
half and half mixture of Hammond organ oil and lighter fluid about 4 oz.
The lighter fluid breaks up the gum on the little oil threads and allowes the
oil to flow freely.
Tried it and 5 minutes later the squeeling was gone forever.
Follow with a good oiling and you will never hear it again. -- Phil A
Often, failure to start can be attributed to a faulty run (or sometimes start) switch.
There is a resistor wired in series with the start motor. When the run switch is in the off position, it shorts out the resistor so that the start motor gets full power. When the run switch is flipped to on (the run motor gets energized, and) the resistor is no longer shorted, so that the start motor gets reduced power, allowing the run motor to "pull" the TG to the correct speed.
At least, that's how it's supposed to work. Often the run switch fails in such a way that it doesn't correctly short the resistor. Then the start motor never gets full power, and can't get the TG spinning fast enough for the run motor to lock. (Solution: new run switch.)
If the start motor is getting the TG up to full speed, and you're sure the start/run switches are okay, it's possible that your run motor is at fault. (Run motors can go bad, but not all that often, so please make to eliminate other possibilites before assuming you need a new run motor.)
Some tests
When the organ is off, try to spin the TG shaft by hand. It should spin freely --- if it doesn't you have gummed up bearings in the TG, run motor and/or vibrato Scanner (or possibly stuck start motor.)
Turn the organ on (without using the start switch). Now the run motor should "hold" the TG and you should feel a fair amount of resistance when you try to turn the TG shaft. If not, either your run motor is not getting power, or it's bad.
The run motor has four leads which go to two separate field coils. Unsolder the leads and use an ohm-meter to check each field coil for continuity. Each coil should read around 180 Ohms, together they are set parallel, around 90 Ohms.
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