Besides cleaning the battery cable connections to the starter motor, were the ground connections from the motor back to the battery cleaned?
This includes making sure the engine mounting bolts are tight.
IF you have a voltmeter, you can do voltage drop tests over various parts of the circuit to isolate the high resistance section.

Model numbers of both the tractor and engine could be helpful here.

Tractor model #256552. Engine is Kohler CV15S. Thanks, I'll check the grounds and mounting bolts.

Are you saying that the starter still makes some noise- like the battery was low or it starts to go and then nothing at all?

When I'm having the problem, every time I turn the key the starter tries to spin, but stops. Sometimes, it keeps cranking normally and the engine starts. It's an intermittent problem.

OP - If you've cleaned all start circuit elect connection points INCLUDING the ground side and engine mounts and are still experiencing the intermittent start problem, suspect problem may be in your starter solenoid - if your tractor is equipped with a separate solenoid bolted to the frame and is of the old Ford motor type.This solenoid has a round internal plate that is pulled down by a coil to make contact for the heavy starter motor current draw.. and that plate rotates randomly, sometimes becoming burnt in spots. When the plate happens to pulls down on the burnt spots, no contact is made. Multiple tries with the ignition switch sometimes cause this plate to rotate to an useable spot - and then the starter turns.
First try jumping across the two large posts ( watch out for sparks) - starter should engage.
Another thing - that solenoid should have either one or two smaller wires coming to it to energize the internal coil.. if only one, then the mounting of the solenoid to the frame is the GROUND for the energizing circuit and must be clean and solid.
Lastly - if the component is suspect, they are not that expensive.. just be sure to get an exact replacement and take before/during/after pixs to keep from getting screwed up on what gets hooked where.
Good luck

99%of the time, if the solenoid is bad, it just clicks and no engine cranking takes place.
It's seldom that it exhibits enough resistance to slow cranking.

Your charge system is only 3 AMPS.
If you don't run the engine long enough to charge the battery well, you may be in a state of always having the battery at low charge?

A totally discharged battery would take 10-12 hours to fully charge with your system.
Maybe you simply need to give it a good charge to get you "over the hump"?
Extended starting/cranking as you describe takes a lot more out of a battery than a quick start.

I keep the battery connected to a Battery Tender device, and have tested the battery so I think it's safe to rule out the battery. I'll check the solenoid and all the electrical connections that everyone has suggested and see if things improve. Thanks.

Did you LOAD TEST the battery?
If not, you haven't tested the battery.

You might also smell the oil in the crankcase for the presence of gasoline.
You may have another contributing issue.

This post was edited by bill_kapaun on Fri, Apr 25, 14 at 12:26