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Their are many such examples in the theory of finite dimensional Banach spaces. Suppose that $X$ is an $n$ dimensional Banach space. If you take a random subspace of dimension $k$, then for some values of $k$ (in particular, if $k$ is of order at most $\log n$), you get a subspace whose norm is a small distortion of a Euclidean norm. In fact, $k$ can even be proportional to $n$ if $X=\ell_p^n$ with $1\le p \le 2$. On the other hand, sometimes you you get a very bad space if $k$ is large. For example, if $X= \ell_p^n$, $2< p \le \infty$, the random subspace of proportional dimensional does not have any good basis (technically, the unconditional constant of any basis is large).