After three days, I discovered that following massive mailings via the API (emails sent in one shot), our dedicated IP had been blacklisted by Microsoft (Hotmail, Live, etc.). After discovering the situation by chance and contacting email support, I was told that our IP had been blocked.
It would be essential that when a majority of emails on a domain are not working, for example due to a blocked IP address, your customer be notified immediately. This alerting system should probably be generalized as soon as abnormal behavior is detected, since you have the information on your end.