Email bombing refers to spammers which constantly hit a lot of email addresses with spam messages.
Their intention is not necessary to bring the email service down, but rather, to load it, or to even if you’re only targeted, to hide your emails, in order to hide the email that states that your password has been reset for example.
More often than not in order for the mail service to restrict users from abusing the service they restrict N space per user, that is for example 1 GB of emails per user, which you have to delete if you’ve reached the limit. Once the limit is met, all of the messages that are sent are going to be automatically rejected from the mail server, and a response that the user’s inbox is full. Hackers can also bombard your mail for this reason.