Help - Network Adapter shows "No Internet Access" but internet works?

So, this is a weird one that I've never run into. I should preface that I'm not a Network Admin, I'm more of a System Admin. I've been with this company for a couple of years and this problem has been around since then but I've typically only ever seen it on Windows 7 but am now seeing it on my own machine which is running Windows 10. The network properties show, "No Internet access" but I'm able to access nearly everything just fine but there's the occasional app that will cause problems, something like OneNote or sometimes Outlook on random machines.

I've tried talking to my Network Admin but he just brushes it off as an anomaly and because most everything still works, he's basically chosen to ignore it. I'm of the opinion there's some underlying networking issue but I don't like to say anything without building significant evidence for a case.

Has anyone seen this problem and have any leads on what might be causing this? One thing that's lit a bit of a fire under the Network Admin's ass is the accounting manager complaining that OneNote isn't working and the no internet access message is a common denominator.

Ninja edit: These are all domain-joined machines. 99% Windows 7, a handful of Windows 10 machines. Mostly up to date, have our standard image applied to them. Issue crops up somewhat randomly.

Thanks in advance.