Knowing who to defer responsibility to is also a vital component in the work place. That's not to say you should avoid responsibility, but if you are part of an organization and a team, therefore it is important not to be personally overbearing for the good of everyone you work with. For example, and this is of course not my speaking from personal experience, but I think that members of a medical team do not have a responsibility to 'save lives'. That's just not the MO. They have a responsibility to try and improve the lives of the living to the best of their ability, but the pressure to be some sort of a hero would be crushing and, ironically, irresponsible as it would probably endanger far more lives than it preserves.
Pressure is often just a variable of exposure. The more comfortable you are with your job and how to handle unexpected situations, the better prepared you will be for them. This almost goes without saying, but it's definitely an important thing to keep in mind at all times.
What do you do for a living, if I may ask?