Yes it can cause upset stomach in some people. It's to do with the Caffeine and tannin in the tea. Try drinking redbush tea (Tetley).
The above reactions to substances in tea, such as caffeine and tannin, are more correctly called sensitivities, as they are exuberant physiologic reactions to these chemicals. Caffeine stimulates several important pathways within the body, particularly the adrenaline-type pathways, which can lead to tremors, fast heart rate and heart palpitations (though generally not life-threatening), as well as nausea, if overconsumed by anyone. In lesser amounts, caffeine can overstimulate sensitive individuals. Tannin, as an acid, can cause GI upset directly.
Allergies are immune system reactions, hence different from sensitivities, that can be manifested in several ways due to the different parts of the system, and can sometimes be life-threatening. The body can develop allergies upon first exposure, or after using a substance safely for years. Larger chemicals tend to incite allergies more easily than small ones. Caffeine and tannin are smaller, so reactions are not common, but possible. As a complex chemical mixture derived from plant leaves, any tea is certainly capable of containing many potential allergens, in addition to caffeine and tannin.
Food allergies can be as simple as an itchy throat after intake, or life-threatening, such as swelling of the face and throat blocking the airway (as seen with shrimp). Skin reactions, such as a widely distributed rash, appearing after a day or so, or wheal and flare reactions within hours, are sometimes seen, but are not common. The most common type of reactions based on food allergy are diarrhea and vague abdominal pain.
If persistent, these can also be signs of immune disease independent of a simple tea or food allergy. See a doctor.
A true caffeine allergy would cause a reaction after any caffeine source was consumed, such as coffee and soda. Different teas contain different mixes of chemical, as well as additives for taste, so one could experiment with different brands and types if one appears to be either sensitive or allergic, though if the latter seems likely, one would need to exercise great care before re-exposure.