If you have spring/summer allergies and a flavored tea makes you ill, you might be reacting to a floral addition to the tea. Those pretty flowers and bits can still cause a reaction even if you're drinking them instead of sniffing them, and some flowers are just a prettier relative of well-known misery inducers like ragweed. (Chamomile, for instance.)
For a regular or unflavored tea that's causing problems, try rinsing the leaves before steeping, to flush any dust from being packed tight and stored. It can't hurt, and it might also improve the flavor.
Completely off the Topic of Tea