To "bomb" or "bombing" an email account - usually referred to as "email bombing" - is the practice of sending a huge amount of email messages to the account so that it can no longer be accessed. Virtually every "paid" or "free" email account has a storage limit; in the past the storage limit of these accounts was much lower: typically 1-10mb, whereas in 2012 it can be 1-10gb.
Therefore, in the past, it was relatively easy to bomb an email with emails - usually including large attachments - that would exceed the storage capacity of 1-10mb; forcing the account administrator to suspend the account. Email bombing is usually classified as a form of a "denial of service" attack, also referred to simple as a DoS attack.
Email service providers do have more armory to protect against email bombing: specifically through the use of spam filters. There are numerous spam filter software application available, and virtually all commercial email service providers have incorporated a filter into their service. Email bombers typically uses the same automated software as email spammers use; however, the techniques used to email bomb are constantly being revised.