Each bottle of wine corresponds to a the set of rats who tasted it. Let $\mathcal{F}$ be the family of the resulting sets. If bottles corresponding to sets $A$ and $B$ are poisoned then $A \cup B$ is the set of dead rats. Therefore we can identify the poisoned bottles as long as for all $A,B,C,D \in \mathcal{F}$ such that $A \cup B = C \cup D$ we have $\{ A, B \} = \{ C, D \}$. Families with this property are called (strongly) union-free and the maximum possible size $f(n) $ of a union free family $\mathcal{F} \subset 2^{ [n] } $ has been studied in extremal combinatorics. In the question context, $f(n)$ is the maximum number of bottles of wine which can be tested by $n$ rats.
In the paper "Union-free Hypergraphs and Probability Theory" Frankl and Furedi show that $$2^{(n-3)/4} \leq f(n) \leq 2^{(n+1)/2}.$$ The proof of the lower bound is algebraic, constructive, and, I think, very elegant. In particular, one can find $2$ poisoned bottles out of $1000$ with $43$ rats.
