Here is a rather vague and subjective question: for which $n$ and $m$ are $\mathbb{R}^n$ and $\mathbb{R}^m$ ``essentially similar''? The answer depends partly on what type of mathematician is answering it.

Of course, in a certain technical sense, $\mathbb{R}^n$ and $\mathbb{R}^m$ are different for any $n\neq m$ since they are not homeomorphic (or linearly isomorphic); thus they by definition differ in some describable topological (or algebraic) way. But qualitatively all Euclidean spaces, especially ones of large dimension, seem to be ``similar.'' The question is really: how large does $n$ have to be for Euclidean spaces of dimension $\geq n$ all to behave essentially the same way?

I have included my own discussion of some possible answers in my Real Analysis manuscript at http://wolfweb.unr.edu/homepage/bruceb/Meas.pdf (Section XI.18.3). I invite comments on this discussion.