Database error when I get high traffic?

I am having some major trouble with my blog. When I hit major peak periods with traffic, I get a “Error establishing a database connection” database. Now, when my site settles to its regular traffic, I don’t get this error. Only at high traffic periods which causes an annoyance for my site visitors as they see this error instead of viewing other pages on my site. It seems to appear randomly when I get the high traffic as I said.

Anyone else experiencing this? I currently host my blog on a dedicated server that has memory of 4GB, so I would assume this would enough to handle the load, right? Or do I need to upgrade?

Any help would be greatly appreciated.