A quick fix for the "A connection to the VAC servers could not be made" or whatever it is
1. Open Task Manager
2. Right click "Steam.exe"
3. Select "Set Priority"
4. Set it to "Above Normal"
I used to get that error quite a bit, and I contacted Steam Support and they told me to do that, and it solved the problem. It was an issue with Steam not having enough time to check the connection, so setting its priority higher made it more important for a connection
You'll probably need to re-set the priority whenever Steam gets opened (if you shut down your computer, or exit Steam etc.)