The first attempt to solve this was to install the latest version of Evernote, 10.8.5, but this didn't help.
I passed some logs to Evernote support and they said I needed to install the latest version, but when I checked, the latest version was the one already installed.
I was then asked to do the following:
Go to the browser interface and check that everything is up to date. (Not easy for me, since I have nearly 2,000 notes. But I've been using it for the last couple of weeks and it seemed to be all there.)
Click the search icon on the Windows task bar (next to the start button) and search for: %appdata%
Hit return or click on what it found, which launches File Explorer in my AppData\Roaming folder.
Delete the folder called Evernote. (I zipped it up first and put it somewhere safe, just in case.)
Launch Evernote - I needed to look up my Evernote username and password to sign in.
It took a few seconds to download all the notes, but it worked.
Looks like this problem was a data migration issue.
