Looked into the FTDI FT232R datasheet.
On pin 26 TEST it says:
Puts the device into IC test mode. Must be tied to GND for normal operation, otherwise the device will appear to fail.
I measured this pin to be open. I soldered the pin to GND (connect Pin26 to 25), tested it.
This is a Gravitech.us Arduino Nano V3.0 with ATMEGA328.
Now I do wonder why a product in its third revision has such a bug.
It is not an unusual use-case to have the device internally powered, and then occasionally connected to the PC, e.g. for a code update.