You MAY have already solved the mystery. "The battery is on its last legs" is certainly one possible reason. If, even when docked, the external power source is having difficulty keeping the battery up to a reasonable voltage, one of the first things that is 'discarded' to conserve power to allow basic navigation to continue are wireless services (cellular and Bluetooth).
However, you could help us by firing up your 540 and navigating a couple of right arrows into the menu system to "Status & Information", then "Version Information", and then the "Network" button at the bottom center.
Please advise "Status" (after the device has had an honest chance to connect), the number of blue signal bars (if they ARE blue) on the left side, and which of the four data fields below are populated with any useful information.