If Thunderbird just quietly stopped getting mail and did nothing everyone would complain they were not told. So what is you specific use case that such a notification is not appropriate to you?

why not give that option to the user. if i wanted no notification i should be able to do it. imagine i have a email account i on a server i don't want anyone to see. and this happened a couple of days ago. i was showing something on my computer to a work collegue, when all of the sudden a popup notification came up. i'm sure he did read what it was. but i didn't wanted to. the other option is to get the email manually. but a simple check box on the settings would solve this.