Yes and no

IINET said it was my network adapter, I said it was the router.  Well, we are both right.

We got a new router from IINET, as our old one packed it in.  Now, I never had a problem connecting to it, or maintaining internet connection with the old one.

This new one, however, is a different story. If I log in, I have internet immediately.  If I start another program that accesses the internet, I will lose internet connection until I start troubleshooter, or just reset my adapter (quicker than going through troubleshooter which ends up doing that).

I had Windows check for driver update for network adapter, and according to Windows no new driver, mine was latest and greatest. Ever the doubter, I went in search of exact make and model of network adapter, and imagine my (non)surprise when I saw there was a new one.  Downloaded, installed reboot, and still the problem persists.

Did a bit more research, and it turns out that some older network adapters (mine, it seems) aren't loved by new routers. 

Solution isn't new network adapter driver (been there, done that, rebooted to prove it, and I have an All in one desktop) but possibly using a newer USB external wireless adapter.  But until I get one, I will do the disabling/re-enabling network adapter whenever I start a new program that access the internet. Oh joy. :p