Why does my console transfer speed suddenly slow down?

With my ISP I pay for 350download and constantly get that speed on my xbox console. So the console does not limit the speed available. It is likely the customer has a wireless connection with many devices connected to the same access point and that can
cause slow down

My download speed is 12 and upload is 1 , but does the slow connection really explain why I can't receive messages?

Well, just to update on the situation. I disconnected my xbox from the internet, waited 5 minutes, reconnected, and I was suddenly getting speeds of around 50-60Mbps again, so the speeds where very high.

It got me to 88% in no time, and then slowed right down again, so disconnected, reconnected and the download was finished within a minute.

No it wouldnt speed up or slow down powering up your console.

Connecting a 100Mbps NIC isn't going to slow your network down. My entire network is gigabit speed, with the exception of a 100Mbps NIC in my network printer and Xbox 360. I still get transfer speeds of over 100MB/s between PCs. 100Mbps is more than enough
for any HD stream.