I hadn't touched this server in weeks, so they generated their invoice, the notification email got filed away as spam, and the server was disconnected (again the message being filed as spam).
It wasn't until almost a week after that I noticed something was up and called them. They advised me to open a ticket, I did, one day later the server disappeared from their control panel, and I have yet to receive a reply on that ticket.
The missing the billing and their messages getting spam filtered is completely on me, but the not answering a ticket and linking ticket-creating-account-ness the payment status of the server is totally on them.
Back to Hetzner, it seems. I didn't lose anything but my blog (which is Octopress and thankfully backed up to Bitbucket) and a couple of VMs I used for learning things.
I've had to stage an emergency exodus of blog content from third party hosts more than once because reasons (most recently: Thanks OVH billing department!), so having content in a non-exportable site scares me a little, especially when I'll be using that site as my primary writing area.