Amongst all this noise some users started complaining of an actual error message, which appeared when viewing Products that had been assigned to an Opportunity. The error message given was:
Precision must be an integer within the allowed range: 0 for integers, 0 to 4 for money, 0 to 10 for decimal, and 0 to 5 for float fields.
After a bit of head scratching and looking for Money fields set to a precision of 5, the problem turned out to be that the Products had not been added to the Price List being used by the Opportunity.
I added the Products into our Price Lists and the problem went away. Of course we had to fire the person resposible for setting up the price lists, we can't let the users think they have won one.