Why do connections to my database fail, but only sometimes?
Is the error you see on that last line always preceded by the same set of jobs, or does the error happen in different places?

I've thought about this for the last half hour and everything I think of I can refute.  Do you have any known networking issues on this server?  I've seen something kind of like this with VMWare's networking stack being flaky and dropping connections.  But then that doesn't make sense because the SQL box is showing an error.

Even though I ask about networking, I can come up with refutations to that, too.  This is a tough one.

If the jobs listed above are the culprit, then I would need to see what they are doing be sure what was going on.