Email bankruptcy is a term used to explain a decision to delete all emails older than a certain date, due to an overwhelming volume of messages. The term is usually attributed to author Lawrence Lessig in 2004, though it can also be attributed to Dr. Sherry Turkle in 2002.
An insurmountable volume or backlog of legitimate messages (e.g. maybe on return from an extended vacation) may also lead to bankruptcy.
During the act of declaring email bankruptcy, a message is usually sent to all senders explaining the problem, that their message has been deleted, and that if their message still requires a response they should resend their message.
References
Email bankruptcy Wikipedia(Text) CC BY-SA