I think I have got to the bottom of the problem. The Beaglebone Ethernet will not work reliably with the Ethernet on my Linux workstation. But it will work with the Ethernet on a couple of other computers which I have. And yet… the Ethernet on my workstation works quite happily with those computers also! The only conclusion is that the my workstation physical Ethernet is a bit marginal and so is this Beaglebone, so they cannot work together. I will swap out the Beaglebone and see if the new one is any better…
Problem fixed. I have fixed the problem and am posting this in case it is useful to anyone else. My workstation has a Gigabit ethernet card. As discussed above it does not appear to work reliably with the Beaglebone. I have found that placing a fast Ethernet switch between the workstation and the Beaglebone completely cures the problem. Of course there should be not problem both sides of the connection SHOULD default to fast. But the Beaglebone obviously does not.
Just for the record, the BeagleBone is 10/100.