The amount of tea you use won't affect the flavor of tea adversely; the way you brew it will. This also depends on what kind of tea you're making (black, oolong, green, white, herbal). Greater quantities of tannins are released in higher-temperature water, which makes the tea more bitter.
TL;DR: If you follow the instructions for brewing temperature and time, you should be able to more or less add as much tea as you want.
Black teas are brewed with boiling water, typically for 2-3 minutes. Green, white and oolong teas are usually brewed at lower temp (180 F) for 3-4 minutes, and herbal teas at just below boiling for anywhere from 4-8 minutes.
I've found that I can easily more than double the amount of tea leaves I use, brew it at the recommended temperature for the recommended time (most tea packages should have instructions), and store the 2x concentrated tea in my refrigerator. Upton Tea has a great guide to tea for water temps and amounts. They also have instructions for hot and cold brewing iced tea - they recommend double or tripling the leaf quantity when hot brewing, then diluting to taste. (I buy a lot of my teas from here, and their instructions have never steered me wrong.)
If you're looking for even more information about preparation of and descriptions / histories of different types of tea, I recommend the Harney & Sons Guide to Tea.