One of my very important, customer-facing servers was acting odd, and I found that when I pinged it, it was giving 40% packet loss. That's the sort of symptom you get on a very congested network; I've experienced that when I was doing some major copying across the network.
So I ran iptraf-ng, a handy utility that shows you what your traffic is; where from and where to. That showed some traffic, but not a lot. If you haven't got iptraf-ng, you can get it using yum.
Actually, I was doing a major copy at the time, but the two servers (from and to) were on a separate segment of the network. That's a good trick, by the way, if you know that two of your servers are going to be passing major traffic, put them both onto a separate ethernet switch. Then that switch might get saturated, but the rest of your network isn't affected.
So, back to the 40% packet loss. I rebooted the server, and that didn't make any difference. So I went down to where the server was, and changed the ethernet cable and port on the switch, and gave the server a cold boot.
One of those three things has cured the problem. And I still don't know what the problem was.