When OOF makes you hit EXO limits and you’re blocked from receiving emails

I recently came across something you might also run into at this time of the year or when a company wide announcement needs to be made via e-mail and the sender is in Exchange Online:

A person sent some season’s greetings, which results into being blocked by EXO as a limit was reached (you can find more about these limits here). My first thought was something like sending e.g.: 10.000 messages per day or too many recipients, but it turned out to be something different.