I have had a DJI Phantom 2 Vision plus until 2 weeks ago.
Sometimes when I tryied to start the propellers they didn´t start and the front red leds flicked. I switched off the Phantom, removed the battery, inserted it again and switched on the Phantom again. Then the propellers started.

With the Phantom 3 Advanced I got not FPV image once and the settings reported 0 volt battery. I followed the same procedure and got FPV image.

I'd like to know if anybody had the same problem. Could it be caused by a bad contact between the data connector of the battery and the Phantom or could it have any other cause?

I think you have hit the nail on the head, it is most likely the data connector. The best way to clean it is with a small rubber pencil eraser.

Or a decent contact cleaner. Some folks even use isopropanol on a QTip.

RedHotPoker

Yeah.. seems  like a contact issue.  Try an eraser and rub against the metal connecting points.