I'm not sure of the specifics for caffeine, but there is an optimum pH for alkaloid precipitation with tannins... I think around 5.4. Black tea has a typical pH a bit higher than 6 from memory. Also, excess tannin can can actually prevent precipitation. Maybe there is too much tannin present in tea to achieve precipitation.

Interesting question. I've seen caffeine/theobromine co-precipitate with cacao procyanidins in organic solvents (e.g., ACN, MeOH). May play somewhat differently in water. Phenomenon likely due to larger degrees of polymerization (DP) in cacao relative to tea. One needs a fairly high concentration of dimeric or trimeric tannin to precipitate BSA. Tea tannin DP doesn't exceed 3DP. Once you exceed