This took me hours to debug yesterday so I thought an article would be a good idea.
My phone was sending email notifications for events in our shared calendar.
Some of my Macs were trying and failing to send similar email notifications. (Mac OS Mail knows about some of my accounts but it doesn't know the passwords because I don't entirely trust the keychain.)
There is no user interface for email notifications in Apple's calendar apps.
It seems that if you have a shared calendar containing an event with an email notification, each of your Apple devices which have this calendar will separately try to send their own duplicate copy of the notification email.
(I dread to think what would happen in an office environment!)
I couldn't get a CalDAV client to talk to Fastmail's CalDAV service in a useful way. I managed to work out what was going on by exporting a .ics file and looking at the contents.
The VEVENT clause saying ACTION:EMAIL was immediately obvious towards the end of the file.
Sadly I don't know of a way to stop them from doing this other than to avoid using email notification on calendar events.