Here are two answers:
(a) If you try to write down an elliptic curve $y^2 = x^3 + a x + b$ with everywhere good reduction, you need to choose $a$ and $b$ such that $4a^3 + 27 b^2 = $ a unit. We can certainly solve this equation over some (lots!) of number fields, say if we set the unit equal to $1$ or $-1$, or a unit in some fixed base number field. But we can't solve it in ${\mathbb Q}$.
[Edit: As Bjorn intimates in his comment below, one has to be a little more careful than I am being here to be sure of good reduction mod primes above 2; the details are left to the interested reader (or, I imagine, can be found in Silverman in the section where he discusses the proof that there are no good reduction elliptic curves over $\mathbb Q$).]
(b) There are many non-trivial everywhere unramified extensions of number fields (e.g. $\mathbb Q(\sqrt{-5}, i)$ over $\mathbb Q(\sqrt{-5})$), but there are no everywhere unramified extensions of the particular number field $\mathbb Q$. The situation with elliptic curves is completely analogous.
