Hi RuthiB,
I had the same situation and wondered about it.
At night I turn of my wifi, but still in the morning I had messages and somtimes even an email.
Then I found it:
It is in the Wifi-settings.
If you click on advanced…
The second setting about scanning is probabely enabled.
This give your apps the right to ‘break trough’ and scan for messages, even if the wfi is turned off.
Hope this helped you.
Michelle