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How can you troubleshoot what is wrong with a broken iPod other than just swap out the parts that you guess are broken based on symptoms? Can you use the diagnostics menu to pinpoint a say bad hard drive or motherboard or bad click wheel etc. Does Apple have a machine which does the diagnostics? Thanks for any info.

There is a set of onboard tools which you can access. Review these guides: How To Enable iPod Diagnostic Mode And Troubleshoot Your iPod & Apple iPod Diagnostic Mode which explains the different tests.