Wireless Question

I was talking to someone from my internet service provider today. In passing, he recommended disconnecting the internet before turning off my laptop at night because he said laptops still send and receive information through their wireless connections even when powered off.

Is this true? I’d never heard that before and am concerned because I’ve not been disconnecting before shutting off the laptop and am not sure how, if he was right, I’d be protected since NIS wouldn’t be running while the laptop was off.