There are two reasons why a mailbox can get full. The most common is that your have hit a limit set by your account administrator. In some cases a mailbox is reported as full because the hosting account has hit the limits of the hosting package.
Both you and your account administrator should be receiving warning messages when your mailbox hits 80% and then 90% of capacity. To avoid loss of incoming mail, it is important that mailbox space is made available as soon as possible.
When a mailbox reaches 100% of capacity, incoming messages are rejected by deferring message delivery. Different mail servers will respond to the deferral in different ways, depending on how the sending mail server is configured. Most servers will retain the message and continue to attempt delivery for some period of time. They will also inform the sender that the delivery attempt has failed. If the message remains undeliverable, then it will eventually be abandoned and returned to the sender. In our case, our server tells the sender when a message is undeliverable for 24 hours and the attempt is abandoned after 72 hours.
In some cases, notably when the message is coming from a web application, bulk mailer, or other automated process, an undeliverable message will simply be abandoned after the first attempt.