I see this occasionally around here, albeit not as much lately. His end customer is likely a Fab for the Semiconductor industry. Internally, most motors used in the machines are servos or on VFDs, so the incoming frequency is irrelevant since they are using power supplies for almost everything (except heating). So to maintain consistency around the world, the Fab owners in North America get custom 480-400V service transformers at the site. That way they use all identical equipment wherever they go. "Copy Exactly" is a term used by Intel for ALL equipment going into any of their facilities, meaning if you supply a piece of equipment to them in one place and they order another one, it must be EXACTLY the same in EVERY detail, right down to the plating of screws (that is an example they use in their training). So as nuts as this sounds, it is the way it is done in that industry.
So back to your problem;
No, you cannot operate a motor that is designed for 480V 60Hz by giving it 400V 60Hz. The motor will still attempt to spin at the same speed (based on getting 60Hz), but the ratio of Volts and Hz is incorrect and the motor will not develop rated torque. Torque varies at the square of the voltage difference, so 400V is 83% of the motor rating, therefore the torque will only be 69% of what the motor is rated for. SOMETIMES an equipment designer will have a "fudge factor" in the motor selection, but not so much any more, and I have never seen more than 20%. So no matter what, your motor will not develop enough torque, which means it will have higher slip, draw more current while doing less work, and either trip the Overload protection or burn up.
When you have a VFD on that motor, you can limit the speed to 83% of rated, which would make the motor see the correct V/Hz ratio and develop rated torque, albeit at the reduced speed. In something like a centrifugal pump that means less flow and may affect the head capability, so it needs careful consideration.
From your last post though, it appears as though the motor is NOT rated for 480V, it is rated for 400V. That's fine, you are good to go. The VFD will just be programmed to put out 50Hz at 400V as per the motor specs and you get full torque and rated flow. It's unlikely that it was ever putting out 480V from the 400V supply, that would not have made sense. That was likely just someone ASSuming something that they didn't understand.