I am a big fan of learning moments. In business, some learning moments come at a cost. Some mistakes are easier to forgive than others. I still believe in exercising as much patience as possible to allow people to learn.
This is easier said than done. Someone makes a mistake (or multiple). Steps need taken to correct course, mentor, and allow the mistake to serve as a positive step for the future. I can't say how many times my bosses or peers have afforded me such opportunities to grow. They demonstrated clear actionable steps, told me what needed to be done to correct the action, and didn't hold a grudge. They were kind. This is a clear demonstration of patience.
It's so easy to throw your hands up, freak out, and make a major issue when someone makes a mistake. Especially in a competitive environment where peers are looking to "one up" others. I especially believe "the easy way out" is to not approach the person directly, but to go directly to their supervisor. This does not establish a relationship with the peer and is a clear sign of trust issues. This trust can help co-workers truly bond as team members, while alleviating petty crap the boss likely doesn't have time for.
Learning the hard way
Be mindful, some people do not respond well to a patient approach. Some stubborn folks require direct mandates from those with whom they report to. I personally find this to be a sign of immaturity and a lack of aspiring for personal growth. It shows signs of a toxic team member. This tendency is often about learning the hard way. Problems get so out of control that they must be escalated to supervisors for immediate action. Watch out for these people.
The high road
At the end of the day, you need to be able to live with your actions. I always encourage that you tread carefully when approaching coworkers about potential issues. Reacting strongly or emotionally out will only make your relationship suffer and cause you to feel worse at the end of the day.
Advocate for patience, promote learning, and be a leader. Ask them if they want feedback. Be thoughtful and highlight the positives if possible. Imagine you are on the receiving end of the feedback. This courteous approach often lowers the defensiveness and can gently open the lines of communication.
Great teams know how to communicate for the betterment of projects or the organization. I believe most people want to grow and become better at their positions. A patient and thoughtful dialog can really enable mentoring between peers and truly promote growth of both the person delivering and the one receiving.