When I open my existing solution in Visual Studio 2012 professional, I got the above error. This used to work but something messed up after the computer is rebuilt.
Soluton for me was to run the standalone TFS installation. This is for 2017, but worked to fix my VS2019.
https://visualstudio.microsoft.com/thank-you-downloading-visual-studio/?sku=TeamExplorer&rel=15
Bit of background...
I'd been having this issue since 2017. Tried all the usual suggestions on here, cache clearing, config file editing etc. but would always have to come off my organisation's network to check code in, as it was proxy related. TFS was trying to connect to a proxy server ip that had been decommissioned, no one from the service desk could work out why. Couldn't find any reference to this old ip in any config files.
Installed VS2019 hoping it would solve the issue but it persisted. Eventually I thought of trying to uninstall it as a component and re-installing it. Couldn't find a way to do that but in my search I found the standalone installer for Team Explorer, albeit for VS2017.
After installing, I ran this, opened up Manage Connections in the Team Explorer connections page and did what I needed to connect to the repository from there. Worked first time. Closed Team Explorer 2017, opened my VS2019, and everything seemed fine in there as well.
Amazed my problem is finally solved!
This error is usually related to the TFS cache. Before you clear the cache folder, please make sure you have closed all instances of Visual Studio and also try to open Task Manager and check if any TFS Services are running. Select each of them and click on End Process Tree.
And also clear the server cache Server cache, the path like C:\ProgramData\Microsoft\Team Foundation\Web Access
Then restart Visual Studio and run as Administrator Mode.
More details you can refer this blog (archived version at the bottom) and also could try the better fix.
Note: Make sure you are working on a different workspace rather then VS2015's.
In my case, my company VPN my Host machine was connected to was preventing me to access the tfs on a VirtualBox VM, returning this exact error message.
This issue is happening when server is not responding, In my case server hosting TFS was down, we restarted and get connected.