In this case, the Task Scheduler status of "Could not start" was caused by the Task Scheduler losing the password of the configured "Run as" user. The Event Viewer logs and the a.logs did not contain any information pointing to authentication or credential problems.
Reentering and then applying the same domain administrator password allows the scheduled tasks to be executed, both manually and by the configured schedule.
NOTE:: As a shortcut, Re-enter the Username and password to the NS.Weekly schedule. When you run the NS.Weekly schedule, it will re-create the rest of the Scheduled tasks with the correct access credentials.
Notification Server 6.0 SP3