What are the pros and cons of doing PhD study at a big vs. small department? Crudely, it seems better to study at a big department - one has more peers, more seminars, and is also more likely to find a faculty member with the same interests if one's interests change during the program. However some top-tier departments appear to prefer to stay small, e.g. Caltech as a whole. That seems to imply there's a downside to becoming a big department, which is part of the reason they haven't expanded (I'm guessing they aren't staying small because they lack the funding - they're top tier after all).

For the student, is it preferable to study at a big or small department? What are the pros and cons of each?
The main disadvantage of being a student in a big department is that the professor can not actually provide in-depth mentorship to that many people: your "real" advisor will most likely be some post-doc instead. The quality of post-doc advice is more variable since they haven't had as long as a professor to prove and improve themselves. (This one bit me personally, and I switched from a largeish department to a small one where I like it much better.)

On the plus side, a big team will likely have several different funding sources, so if your main grant runs out but you don't have enough work to graduate yet, it'll be easier to find you another contract to tide you over.