Q:

Potentially good, semi-stable reduction => good reduction ?

Does a smooth proper variety having semi-stable reduction as well as potentially good reduction have good reduction ?
Note that over a $p$-adic field, this is true for the Galois representations in the $p$-adic étale cohomology of $X$.
(With a bit more details: fix a field $K$ complete for a discrete valuation, with ring of integers $\mathcal{O}_K$, and a smooth proper $K$-scheme $X$. We say that $X$ has good reduction if it is the generic fibre of a smooth proper $\mathcal{O}_K$-scheme $\mathcal{X}$. We say that $X$ has semi-stable reduction if it is the generic fibre of a flat proper $\mathcal{O}_K$-scheme $\mathcal{X}$ such that étale-locally on $\mathcal{X}$ there exists a smooth morphism $\mathcal{X}\to \text{Spec}(\mathcal{O}_K[T_1,\dots,T_r]/(T_1\dots T_r-\pi))$ where $\pi$ is a prime element of $\mathcal{O}_K$. We say that $X$ has potentially one of the two properties above if it has it after a finite extension $K'/K$.)

A:

No. Take $K=\mathbb Q_3$. Consider the projective genus $0$ curve $x^2+y^2+3z^2$. This has bad reduction, since it has no rational points. It is semistable, since after adjoining $i$ it has exactly that form. It has potentially good reduction, since all genus $0$ curves do. 
So you at least need to say it has good reduction etale locally.

