By default, your SMS text messages will be stored for 60 days (from the time the message was sync'd to MightyText).

You can change this time frame in the settings page of the MightyText Web App.

The options are:

Within 24 hours of this setting, your messages and your call logs will be removed from MightyText.

*This setting does not apply for your photo MMS messages.  They will not be removed and are stored indefinitely.