Ten tasters finding one poisoned wine barrel

Exactly one of $n$ barrels of wine is poisoned. Ten volunteers can taste wine; anyone who drinks poison dies exactly one month later and otherwise survives, and there is only time for a single round of tasting lasting that month. What is the largest $n$ for which the poisoned barrel can always be identified?

$1024$. Each volunteer's fate is one bit, so ten volunteers give $2^{10}$ distinguishable outcomes. Label barrels $0$ to $1023$ in binary and have volunteer $k$ sip every barrel whose $k$-th bit is $1$; the set of deaths spells out the poisoned label.