They are funny places. There is no standard for how they are run and what works for one person could easily be considered awful to another. That said, there are a few generalities for how things will work.
PI: The principle investigator is (usually) the professor in charge of the lab. He/she can be identified on journal articles as being an author to whom correspondence should be addressed (typically named last). The PI is the idea person. PI's, of course, do lots else: teach (generally), write papers, write grant proposals, give seminars/lectures, sit on committees (for doctoral students), and generally guide the direction of the lab. Typically these all go hand in hand and the best PIs are good at all of them. Universities tend to care about grants first and teaching a distant second (at smaller schools, not so much).
Grad students: Graduate students are the backbone of the lab. They know the techniques, who/where to go (to). They are the experts within a lab. They are there longer, and generally have at least one year of training before anything is really expected of them. They may take time to find what they like doing best, and they may drift through several projects, but they are where the work comes from.
Post-docs: Post doctoral associates are a bit of an oddity. Their education makes them senior, and they are typically experts in a field, but they are not (usually) the most capable lab members. This is not because they are poor technicians or that they can't learn, but because that is seldom their focus. Post-docs are stuck in between PI and grad student. Under the best of conditions a post-doc will come into a project that has others working on it and will provide some new idea/ability. Maybe they will have better ideas on how to analyze a molecule/material. Maybe they will know a technique that is well suited to a problem. Maybe they will have a background that makes them better able to judge a technique and determine what should be done to improve it. Utilizing a post-doc's strengths is, in large part, the responsibility of the PI. Failure to understand what those strengths are, is the fault of the post-doc.
Obviously post-docs are problematic. But even more so than was mentioned. In chemistry most post-docs are around for two years, three tops. They are often interested in becoming PIs on their own at other universities and have to spend a large amount of time preparing for that (research proposals, teaching philosophy, grant searching, ...). This also means, however, that they have research goals which they wish to pursue independently. This creates problems. In graduate school, much of the time a student spends thinking about research, they are thinking about their research in the lab. A post-doc spends some portion of that time thinking about their potential future research. The time frame for applying means that a PI is not likely to get nearly as much dedication from a post-doc as a grad student (this is financially sort of fair as post-docs are often cheaper than grad students because of the no tuition thing). Post-docs that come into a situation in which they are expected to pursue research largely independently are not in a good spot. Things become worse if the research (techniques, background, etc.) is fairly new to them, i.e. not what they did as grad students.
In fact the primary motivation for productivity as a post-doc is that it is helpful/necessary to get that next job. In the face of things not going well: bad results, highly tuned techniques, little in the way of bench assistance (provided or offered), motivation ebbs. ...I would say that the primary motivation should be learning, but I've been told that I'm somewhat of an idealist in this regard and that universities prefer lock-step productivity to idealistic intellectualism (ignoring the fact that it is the latter that provides society with its greatest breakthroughs and achievements).