When one has had so much to drink that they are claiming they could drink forever and that there is no such thing as too much rum.
The phrase plays on the popularity of rum because of rum's sweet flavour as well as it's feature in Pirates of the Caribbean.
This phrase is especially useful by pirates.
Guy 2: Not enough.
Guy 1: Seriously you've had too much.
Guy 2: There's no such thing as too much rum!