Will a USB cable with no 5V still work?

ReverseEMF:
I took a look at the Nano Schematic, and Yes, it will work. The FT232RL IC that converts USB to RS232TTL is powered by the +5 bus, which is supplied by, both, the USB "VBUS" line (from the USB connection) through a protection diode, AND by the Linear Regular that drops Vin to 5V. It's unclear whether the NANO can be supplied by a direct connection to the +5V output [the Arduino Uno can, but the circuit is a little different]. It depends on whether or not the LM1117IMPX-5.0 IC can survive such a connection.

So, as long as +5V is supplied externally, then no need for the USB powerline.

Ah thanks! And yes... the 5V pin will survive external power. BTW... can you share a link to the NANO schematic?