After the timeouts, there were actually moments where I couldn't connect to my server at all. Restarting the OpenVPN connection alleviated that problem.
So checking the log, I found that the openvpn reconnected when it timed out, and I saw this message:
Apr 3 01:14:15 corinth ovpn-server: MULTI: new connection by client 'ghostwheel' will cause previous active sessions by this client to be dropped. Remember to use the --duplicate-cn option if you want multiple clients using the same certificate or username to concurrently connect.
Aha! What IP was I coming in from?
Apr 3 01:14:15 corinth ovpn-server: MULTI: Learn: 172.16.0.6 -> ghostwheel/18.104.22.168:10654
And that resolves to an DTAG hostname:
# host 22.214.171.124
126.96.36.199.in-addr.arpa domain name pointer tmo-098-92.customers.d1-online.com.
So the solution of the problem was after all quite simple: Another machine was basically kicking me out of my VPN. And yes, I remembered that my girlfriend borrowed my Samsung NC 10 - and she probably left it running when she went to work. I hadn't remembered that I had just copied the VPN config including certificate when I set up my home PC, but of course it was easy to fix.