I agree with a lot of the points that others are making.
I find that low-quality tea (which is not the same as low-price tea) often makes me feel rather crummy, especially if I drink a lot of it. Also, improperly brewed green tea (with too high a brewing temperature) often makes me feel pretty bad. High-quality oolong is probably the type of tea that I can take the most of before I feel any unpleasant effects. But too much of anything can be bad.
With oolong tea, I notice that a single cup tends to be milder on my body than black or green tea of similar-tasting strength. However, if I sit there and make many infusions of an oolong, it can get to be too much.
I think the important thing is to listen to your body. Different people have completely different tolerances to caffeine, as well as to the acidity of tea (all tea is somewhat acidic), and probably to the other chemicals in the tea. It likely depends on what you've eaten and other factors in the tea.
I've found that there's no general rule about how much tea is too much. From my experience, there are some times when I don't even feel like drinking a single cup of tea, whereas there are other times where I can happily drink cup after cup with no adverse effects. It's usually the caffeine that eventually gets to me, but other times my stomach feels off before I notice any adverse effects from the caffeine.
Just stop drinking it if you start feeling like it's too much.