I have also not seen this message, Winston Churchill. That is an interesting observation.
Can you tell us more about your iCloud Photo Library?
How are you connected to the Internet? Is it Wi-Fi, Ethernet, or a cellular Hotspot?
Are you using third-party security software? The reason I am asking is that some OPs reported that their library will not sync over cellular hotspot, and also, that they had to disable their third-party firewall.
Is your photos library on an external drive, that may occasionally become disconnected? If a drive occasionally dismounts itself or becomes unresponsive, it may explain the pausing, but not the "Not connected to the Internet" message, but sometimes timeout errors and "not connected to the internet" may be caused by other problems.