"Have you tried turning it off and on again?"
This may save someone searching on Google from going mad...
So after running OpenVPN Connect on Mac OS X for some time, I went to add a new VPN connection via "Import From local file..."
Existing VPN connections were working fine, but every time I tried adding a new profile I got the error message:
"Profile Import Error"
So I tried uninstalling and then re-installing a previous profile that was currently working. That gave me the same "Profile Import Error". Apparently the Import process was broken.
Restarting my Mac didn't help.
In order to get everything working again, I had to Quit OpenVPN Connect, delete OpenVPN from the Application folder, and then re-install OpenVPN Connect using the installer.
When I re-imported all of my profiles, all of them imported correctly. Apparently OpenVPN Connect can't recover nicely from file corruption...