There was an old thread about this being caused by the official kindle cover shorting the LED booklight power contacts in the Kindle 3 side slots that the cover snaps into. Amazon had a replacement cover to fix this problem.
If you have an original amazon cover, try removing it to see if the problem goes away. If so, get a replacement for the defective cover from amazon.
EDIT: I see that the problem went away, so perhaps not a cover issue, but still good to know...
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