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<p>I guess it makes sense to stay at the same university if that option compares favorably to most other alternatives you can think of, for example if you are an engineering student at MIT. Otherwise, I think it’s better to move on. </p>
<p>Most people usually tend to look for a grad school that is ranked higher than their current undergraduate college. In graduate school though, sometimes the person (advisor) you work with matters more than the overall reputation of the school you attend. It might be advantageous then to go to a marginally lower ranked school as long as you get to work with that particular top expert in your intended field of study.</p>