Sometimes when things don't work out between a student and advisor (no matter who is at fault, if anyone), the student may try working with another advisor in the same department*. This situation typically involves graduate students, but can also involve undergraduate research students. In some cases this is a neutral situation in which the student and advisor mutually and amicably 'divorce'. In other cases, however, this parting-of-the-ways creates a tense situation for at least some of the individuals involved: the student, the former advisor, and/or the new advisor.
The most tense situations are when the first advisor feels that the student has been given ample opportunity to show their abilities, and changing advisors isn't just a matter of exploring new research interests, it's shopping around for an easier project. If the advisor feels the student has demonstrated a lack of ability for Ph.D. research, for example, he or she might be annoyed if another advisor doesn't respect that opinion and agrees to advise the student. Of course the student's point of view might be that the first advisor or research project was deficient in some way, and so they want to try working with another advisor.
The very first graduate student I advised started working with me through an amicable arrangement between her former advisor and me. When I showed up in my first tenure-track position, this student was just starting her second year of the Ph.D. program. She'd spent most of the first year taking classes (and doing well in them) and thinking about what her research focus would be. Her interests matched mine well, and it was fine with her first-year advisor that she work with me. This was an ideal situation. In fact, I think it is by far the best 'inheritance' situation in which I've participated.
The second graduate student I advised started working with me through an amicable arrangement between his former advisor and me. The student said he couldn't get interested in his first advisor's research, but said he was very interested in my work. I took him on as an MS student, though it quickly became apparent that he wasn't interested in much of anything involving science or research. After trying, and failing, to work with me**, he said that he would probably do better if he had an advisor who had the same religious-ethnic background as he did. Remarkably, the graduate advisor allowed him to switch advisors again, but the student continued to fail and eventually left graduate school.
In both cases, the inherited student proclaimed an interest in my research field, but only one was sincere about it. Could I have figured that out and saved myself the stress, lost time, and wasted resources of Student #2? Unless we want to start giving lie detector tests to students, I think not. And even if we did have Research Interest Lie Detector Tests, they would be unlikely to work, as I think that some students may sincerely think they are interested in something, when in fact they know nothing about the topic and are just being hopeful or deluded.
I've also had several other students decide that they would rather work with another advisor than with me. In most cases, the students were failing and in danger of being forced to leave the graduate program. In these cases, no other faculty were willing to advise the students and the students had to leave. In at least one case, however, a failing student was allowed to switch advisors. In that case, the student didn't do any better with his new advisor than he did with me, and had to leave the grad program.
If there is any indication that a student is doing well or could do well with a change of academic scenery (e.g., the advisor and research project), a department has the responsibility to do what it can to help the student succeed. Certainly there are evil and/or negligent advisors, and students should be given assistance in those cases, including the option of changing advisors. If the student is failing or being unproductive, however, in the vast majority of cases this means that the student won't do well with any advisor. I've heard students say "If only I had a more interesting project, I would be more motivated and then I would do better", but in most cases that is delusional thinking.
There may be situations in which a student is locked into a specific project and has to do boring tasks A, B, and C, and nothing more. In that case, perhaps the answer is a change of projects/advisors. In most cases with which I am aware, however, the students have the opportunity to take some initiative with their research and make it interesting to them. If they don't -- and continue complaining -- then it may be an indication that it's not the research project and the advisor that aren't right for them, it's research in general.
* See also Firing Your Advisor (Jan 2007 post)
** This is the student I have mentioned before who, in his first year of graduate school, asked me "Why do you seem to think you know more than I do?" (in reference to my research specialty)
7 months ago