New software development efforts should go through 3 phases of testing before the software is introduced into a real, production environment:

Phase 1: Unit Testing - This phase of testing is performed by the software developer.  As the developer writes new code to perform new actions, he/she must perform functional tests to verify the code they've written works.

Phase 2: Quality Assurance (QA) Testing - This phase of testing is performed by a quality assurance engineer.  The QA engineer is responsible for testing the code using a set of real-world scenarios.  The QA engineer also attempts many different combinations of actions against the software to expose weaknesses and data validation error in the application.

Phase 3: User Acceptance Testing - This phase of testing provides a "sandbox" for real users who will be using the software to do their job.  These users verify the quality of the software and provide feedback on the usability of the tool and whether or not they are able to perform the required actions.

Throughout the testing phases, the goal is NOT to determine all the things the software doesn't do.  Moreover, the goal is to determine if the software does what it is supposed to do.