Everything is "critical priority" all the time. Every new project is the most important project in the entire company. Every request that comes in has to be handled immediately. I have a good manager now who fights back against the deluge of critical work, but for my first year in my job I had a different manager who would bend over backwards to appease everybody, over-promising constantly.

I eventually started asking questions like "Which project are we de-prioritizing to accommodate this?" or "Is X more or less important than Y?" and then I would focus entirely on whichever project he identified as being the most important, and not touch anything else until I was done. Basically forcing him to prioritize our work.

I almost quit over a few of these issues, but I stuck it out and eventually our team came under new management, and now our manager is the one asking those questions instead of me. As she should be. Her favorite response when someone says a task is critical is "How critical? How much money will the company lose per day if this is late?"

Most of the time, the answer is somewhere in the range of "nothing" until a couple months after the deadline. So we set a much later deadline and get the work done right.