I use the lighted cover, and bend it all the way back to read one handed, and have never had a problem. Furthermore the Amazon page for the cover has a picture that shows a model using it this way (not one of the customer-submitted pictures; this is one of the pictures Amazon put up in the first place.) So it's intended to be used this way, and you aren't using it wrong or anything.
I agree with Harry--the problem is probably with the contacts inside the Kindle. I think you should take Amazon up on their offer to send a new one. They are very good at making this convenient and as painless as possible.
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