I think random graphs show that this parameter can be at least as large as $n/ \log n$.  As is often the case there are some details to be checked, but I'd try the following.

Let $G \sim G_{n,1/2}$.  Then the clique and independence number of $G$ are both around $2\log_2 n$ with high probability, and the chromatic number is around $n / 2\log_2 n$.  Bollobás's proof of this fact goes by showing that the independence number of $G$ remains around $2\log_2 n$ even after we've already removed a large number of independent sets of that size.  So I expect $G$ to contain about $n / 2\log_2 n$ disjoint maximal cliques, giving $|T| \geq n / 2\log_2 n$.  (And $|A| \leq 2\log_2 n$ as are there are no larger independent sets in $G$.)

The main thing to check is that the maximal cliques don't conspire to share vertices, but that seems unlikely.