Basic sanity of compilation and basic functionality can be automated, but often things like complex usage, race conditions, network conditions, or tools that interoperate with other components cannot be accurately tested.

Thus, automated (aka computer-run) testing helps the programmer but it can never assure perfection in the code in non-trivial cases. Use caution. Basic tests are a good idea, but it is important to fully understand what either (A) your tests don't cover yet or (B) what it is infeasible for you to automate. Use caution especially around XP advocates, who are known for writing tests such as "verify 1 + 1 = 2". Ok, I exaggerate, but testing for seeming tautologies (while sometimes important), does not good testing make.

I still, to this day, prefer reading code, understanding it, and testing by hand. That may be because I work with pathologically untestable software, but it's also based on a need for caution and a higher level of testing.