The way this site is hosted is through a VPS (Virtual Private Server) service. There is a single beefy system that partitions the resources between several customers. Within each VPS container, a customer is free to sub-divide those resources to one or more hosting account. I'm even allowed to over-provision my allotment. For example, I get allocated a certain amount of CPU, RAM, Disk, and bandwidth. When I say that there are multiple sites hosted, it is a slightly loaded statement. From my resources I'm hosting 5-6 websites. On the physical system as a whole, it could be much more than this. I have no insight to how many are on this server. It could be possible the server has been out-grown.
I chose the VPS because it was easier to manage SF with root access in my container. Their service is a managed VPS so I can go to them for help when I get stuck. The database backups are larger than most shared hosting providers can mange through their automation scripts (they typically time out).
I'd like to figure this out so that it stops interrupting the small number of us who visit regularly. I'd even prefer to have a dedicated server but it's just not economical to do so. I appreciate you mentioning the issue rather than just living with it.