Sure enough, following Mike's advice I stuck an old 3Com 10/100 NIC in the box, and the dropped packets problem vanished.
There really should be a health warning on the NASLite home page regarding the Realtek NIC chipsets. The real problem, of course, is that the Realtek chips now seem to be everywhere. Some of the recent Intel boards even seem to use them.
From Googling Linux forums, people seem to get around the issue by compiling their own kernels including drivers downloaded from the Realtek web site. Life shouldn't be that difficult. And of course it's not an option with NASLite. Now I have an Atom board with one PCI slot and I have a choice between using a decent NIC card or the SATA PCI card I need to support all my HDDs...
On top of that...the problems with my new box have not ended. It's still disappearing from my network on a regular basis. I can't even ping it. The box might stay up for a minute or a day, but it eventually vanishes from the network -- once or twice I have seen an error message similar to the one in
this thread. As well as the NIC I've also replaced the network switch (replaced 3Com 10/100 switch with an Allied Telesis gigabit box) but that hasn't helped. Bad memory? Motherboard? Power supply? Who knows. Running MemTest as I write this. I must have got lucky with my first NASLite box. Put it together with parts that cost £20, stuck it in a cupboard for years and it just worked...