Five tasters, three months of poison testing

Exactly one of $n$ kegs is poisoned. Five servants volunteer to taste for a period of $3$ months; anyone who drinks from the poisoned keg dies exactly one month after drinking, and safe kegs are harmless. What is the largest $n$ for which the king can always identify the poisoned keg?

1024. Each servant now has four possible fates: dies after a month-1, month-2 or month-3 tasting, or survives. Write keg labels in base $4$ with five digits; digit $r>0$ for servant $k$ means servant $k$ tastes that keg in month $r$. The pattern of deaths and their timing decodes the keg, giving $4^5=1024$.