On oak there were some filesystem errors, sufficient that the system wouldn't mount the root filesystem. Booted from a CD, fscked it, and then it was fine.
On willow there were I/O errors. I opened it up to look at the cable which seemed to be well attached at both ends. I took it off and reattached it. Then all was well. I think maybe the cables that came with these motherboards are just poor. I don't know if gluing them will help - the ends were attached, but obviously somehow the connection had become dodgy.
If anyone gets the chance, pick up some new SATA cables. Six would cover these machines and the server too, and we need two ATX-SATA power adapters for the server as well.