I just discovered that my team's EV3 rechargeable battery is defective.
Apparently this is a common problem, but I wasn't aware of it until now!
See: http://www.firstlegoleague.org/sites/default/files/LEGO/Message%20to%20Customers.pdf

If you're having mysterious problems with the EV3 batteries not
holding a charge and the EV3 not turning on after charging, that's
most likely your problem.

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