OK, well I am noticing I get this problem even after religiously signing off, so must be some google quirk...

It could be real, not a google quirk.

Your ISP gives you one IP address. All of the devices on your network share that IP address. You have a router between you and the rest of the world that makes this work. The router remembers your devices and translates the requests between 192.168.x.x and the IP address your ISP gives you. This is called NAT, network address translation.

There are a lot of reasons why it's done this way, some better than others. :-) The effect is that your home network is hidden from the rest of the Internet. Google cannot identify an individual machine on your network, it can only identify your router.

To prove this, type in http://whatismyip.org/ on any device on your network. You will get the same IP address back. That's the address the rest of the world receives from your router.

So the message from Google may be telling you that some other device on you network is checking email. It could be a computer, a phone, a tablet, whatever.

Excellent point and easy to test -- shut off all devices that use that router then turn one back on and see what gmail says.

Also, you should be able to click the "details" link at the bottom of Gmail to see the other IP activity and a button to "Sign out all other sessions".

Try clicking that button and then what does Gmail say?