Here is my analysis for a theoretical subject like Math:
(Also assume I am super-smart which I am not):

1. I assume the stipend would be the same. So will the facility.




2. There is a change that a good university has a more accomplished professor. But since this is research, I can always go and meet him. Also, many schools allow having a guide from outside the university.




3. From my experience, the elite university always has a difficult curriculum along with some really book-smart students. So I would be really tensed to pass the exam and hence my research would suffer (Note here I am making a distinction between book-smart students and research smart students which in my opinion are not always same).




4. University brand is a definite thing. I don't know how crucial this would be after PhD. I for sure know that for undergrad, getting a high paying job was directly proportional to the brand of the university he/she graduated from.




5. Less pressure as people would assume I am not smart (because of mid-level university) and hence I could excel without any performance pressure except for the one which is required to keep the stipend going.

I would like to have some feedback from the current grad student regarding the validity of my reasoning. Feel free to add your own explanation as well.

NOTE: Not so good doesn't mean a bad university, it just means a mid-table university and good university means universities like MIT, Caltech etc.

This post is far too broad to be answered, but I can give some anecdotes.

In my experience, there are no "bad" universities provided that you go to a university that has some kind of national standing. Sure, you can go to the Back Alley University of Cat Medicine and Mathematics, but I don't think that's what you're getting at. More likely you're talking about whether you should go to a university that has national or international standing (like MIT) compared to one that might have decent regional standing but that isn't a national research powerhouse in your field.

Where you do your grad studies should depend on many things. The status of the university is perhaps one of those things but is certainly not at the top of the list, especially for PhD. You should instead be looking at individual researchers. Do they do the research you are interested in? Do they regularly publish? Regularly attend conferences? Then go and meet with them or speak to them via email about mutually relevant research opportunities. Determine if this is someone who can teach you to become a good researcher in the field.

Your research credibility depends on the problems you solve and the contributions you make. An excellent school does not make an excellent researcher just by virtue of having them attend the school.

So, what does a top tier university really offer? First, it tends to attract top tier talent and lots of funding to go along with it. Second, it is highly competitive, so you will be surrounded by other students of presumably high quality. Third, yes, you get a bit of the brand to go with it.

What does a smaller university really offer? Assuming you chose your advisor right (hard problem!) then you will probably get the same quality or higher training to become a researcher yourself, compared to going to a big school. The problems you are working on might not be ground breaking but will always be interesting and relevant - research doesn't care where you study, it cares what you do.

Finally, it depends so strongly on your field which university is the best. Speaking about my home province of Ontario, Canada, we certainly have some universities that are considered to be top-tier, probably world class, but only in certain fields. There is a smaller (but still highly regarded) university whose past is rooted in agricultural research and, despite being a smaller university is extremely highly regarded in that field, probably best in the province for it. However, it is certainly not a world class hub in, say, human computer interaction. They wouldn't compete with MIT's media lab there, but that's not to say that someone interested in HCI couldn't make a huge impact going there.

UK perspective

In the UK, the most important factor for a PhD is the supervisor, not the the institution. Unlike undergraduate or master's courses, a PhD does not involve any "classes", so the role of a department is limited to providing a supportive environment for research. The curriculum for PhD study is essentially the same everywhere -- you write a thesis that makes an original contribution to scholarship (of course, there are same practice-based programmes where other types of output may be admissible, but these are usually a different degree title, such as DLitt, DMus, &c.).

Of course, great supervisors are not necessarily distributed equally among institutions -- often, a particular institution may be notable for research in a particular field, causing the best researchers to cluster there. This can vary enormously by field or even sub-field. At the same time, there is no official league table nor a centralised system for allocating academics to institutions, the result being that some of the best supervisors may not be in the obvious places, whether because they are undervalued or because they prefer to work there.

So, to cut a long story short, you should be considering the different between a "good" and "not so good" supervisor.