$s, t$ propositional terms;
If $s$, $\neg t \vDash t$ and $s, t \vDash \neg t$, then is $s$ satisfiable?
Now, surely there is no valuation such that $s$ and $\neg t$ are true, then $t$ is true? But this is just because of the $t$ term - am I right in saying $s$ could be anything?
In such a situation, $s$ is unsatisfiable.
One way to see this is to recall the interaction between entailment and implication: $$A, p\models q\quad\iff\quad A\models p\rightarrow q.$$
This is an immediate consequence of the semantics for "$\rightarrow$." (Incidentally, when we look at deduction instead of entailment the corresponding fact is the deduction theorem.)
Now we can rewrite your hypotheses as $$s\models \neg t\rightarrow t\quad\mbox{and}\quad s\models t\rightarrow \neg t.$$ Putting these together, we get $$s\models t\leftrightarrow\neg t,$$ so $s$ is impossible (that is, unsatisfiable). A hint for this: if $\nu$ were a valuation making $s$ true, then what can $\nu$ do about $t$?
What's going on here is that $t$ is the way we see that $s$ is problematic. But the problem is really $s$ by itself, which is somehow letting us go from $t$ to $\neg t$ and from $\neg t$ to $t$, and nothing satisfiable can do that.