Perhaps harsh survival advice, but It is not worth hurting yourself over the failure of others. If someone else makes a mistake, why give yourself a heart attack? It is not your fault, so do not pretend like it. And it’s not your responsibility to clean up another person’s mess, whether you know how to or not. So relax; offer a helping hand, and take comfort in the knowledge that you are doing everything that you can to help. Do not stress.
If his or her failure effects you or makes your job harder, it does not resolve the issue to get upset. Anger, frustration, yelling, pouting, and blaming do not smooth things out at all. Stress is almost as unhealthy and contagious as the flu. Your stress will snowball into other people’s stress, which will only come back to hurt you more and make the situation worse. Objectify the problem, take new obstacles at face value, forgive the mistake, and move forward. Do not stress.
If the mistake was made under your leadership, then it is your responsibility. That is one of the risks of leadership, and you need to be prepared to handle the failures of your staff. That said, it is also your responsibility to facilitate recovery from said mistakes. Losing your cool will lose you the upper ground over your problems. Stay focused, think through the situation objectively, realign resources as needed, and act. Decision making is math, not drama. Do not stress.