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You don't want to hear it... but first before troubleshooting anything else, charge the battery fully. Then, is the phone locked to ATT (while that won't cause the reboot issue, it will be asked)... look at How do I determine if my BlackBerry is "unlocked"? - BlackBerryFAQ

If the device is plugged into your PC, it is receiving a charge, and this powered, hence no problem... makes me think... well, you asked us not to say it.

Also, when the radio signal goes on, it does begin to use more juice, reboot, bam, which makes me think... oh well, again, I can't say it.

But I think you get my drift? Charge the battery fully for several hours to 100%, and check back in here afterwards.