Can graduate students be forced to leave graduate school under pressure from their advisers? And how often does it happen? I know someone who was de facto kicked out of Harvard due to some conflict with his advisers, though he was later able to transfer to Caltech.

As an example - what if there was an irreconcilable difference between the student and adviser, and what if other advisers were reluctant to take the student on (for whatever reason)?
This is such a loaded question I'm hesitant to answer. While it might seem that a student was "kicked out" of a program for conflicts with an advisor, most departments (including ours!) have a procedure and policy for when students are asked to leave the program. 

Usually, the reasons would be some mixture of lack of basic minimum grades and lack of satisfactory progress. if there's conflict between a student and advisor, there's usually some departmental mediator (a director of the graduate program) who should be able to step in and deal with the situation (either finding the student another advisor, or something like that). 

But I'm not aware of it being generally possible for an advisor to fire a student and have that student then be removed from the graduate program: these two things are usually separate.