Those errors can happen when a script fails to establish a connection to a mysql server, thus failing to generate a successful resource (which makes all subsequent scripts calling that resource fail and generate warnings). This can be because of scripting errors (which is unlikely in this case because with a scripting error it will NEVER work until the script is manually fixed) or a server that can glitch out and stop responding to requests or just stop working in general. This likely means that the server that stores the SQL database has gone offline, which is why the outage typically only lasts for a few minutes at a time (perhaps a reboot). Aside from changing server providers, there's not a lot that TCP could do to fix it.
Edit: Also, given the max_user_connections error, it could be that too many people are trying to load the site at the same time. I'm not sure the max user setting is something TCP has direct control over, though.
