For any system that matters, you have to prevent a single (or multiple) failures from sinking it. Any new failure that happens should be evaluated in a blameless manner and solved by improving the system so mistakes of that class cannot happen again.
Hire people (on good money) with the personality type that likes to find problems. If the team is all composed of "Mr Right-Now" and no "Mr Right" personality types then it will move very fast and have lots of issues. A diverse team helps.
TDD tries to insist tests are developed first, partly to ensure they're not skipped.
Reinforce the message: "Measure twice, cut once".
Good leadership and top-down support to make changes are important. If they won't care enough to stump up the money and follow through to check it's happening, don't expect much change.