We all make mistakes -- every single one of us. From the big to the small. Of course, sometimes what we think is small is actually big, and what could be grounds for termination is actually not -- it all depends on perspective and situation.
For people with integrity - professionals who work hard every day to do their best and make it happen -mistakes are crushing. They can be earth-shattering. As a manager, this is what you want -- to have the people who work for you feel and react to their mistakes with horror - knowing there is no excuse. There is nothing you can do to make them feel worse. Nothing you can say to make it OK for these people. If you fired them, they'd think it was justified.
BUT -- mistakes very infrequently made by good people who work hard -- are not the problem. It's the repeated mistake made by people who don't care - who continuously cut corners, or leave their teammates hanging, or don't actually bother to try -- that you need to worry about. Those mistakes aren't excusable either -- but they are more predictable. Any predicted mistakes are YOURS, not theirs.
In the end, you don't want mistakes to define your staff -- I think you want how YOU react to them to define you and your business.