Ten testers, one month, one poisoned barrel

A king has $n$ barrels of wine, exactly one poisoned, and $10$ servants willing to taste. Anyone who drinks poisoned wine dies exactly one month later; otherwise they are fine. The servants agree to only one month of tasting. What is the largest $n$ for which the king can always pinpoint the poisoned barrel?

1024. Number the barrels $0$ to $n-1$ in $10$-bit binary and let servant $k$ drink from every barrel whose bit $k$ is $1$. The set of dead servants spells out the poisoned barrel's number. Only $2^{10}$ death patterns exist, so more than $1024$ barrels cannot always be separated.