Many possibilities here, but from the few real details we have, if you mean OFF as in powered down, I would say that the network card is lowering the voltage of the net data across the switch.
(causing packet loss, but you can test that with a ping inside your nat)
Try a different router/switch or different net card or even swap the switch ports around. Sounds like a cheap router/switch and or cheap nic or both.
You might also look in the BIOS for utility functions of the network while asleep etc.
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