I would show the error to your host as they may have a very small amount of simultaneous connections allowed for the same user and that would cause the intermittent error you are seeing.
We are using HostGator and I have spoken with two techs already. Both of them do not currently see the issue and therefor can not help. The error appeared yesterday afternoon. Then a couple days prior to that.
I'm not sure they need to see the error as it should be pretty self-explanatory for a hosting tech. I haven't seen this come up recently on the forum and I believe there are a handful of Hostgator users here.