If I add four spoonfuls of powdered ice tea to a 1/2 liter glass, but it doesn't dissolve, would adding more water dissolve it?

If you add four spoonfuls of powdered ice tea to a ½ liter glass, and it does not dissolve, then it can be said, in terms of solubility, that the solution is saturated. The solubility is the amount of a substance, or the solute, that dissolves in a given quantity of solvent (in this case water) to produce a saturated solution. Because  the solution is saturated, the maximum amount of solute (the powdered ice tea) is dissolved in the solvent.

One way to think about it is that it is similar to when an auditorium is filled to its maximum capacity. No more people can be brought in safely to sit in chairs.

If you were to add more water, then yes, the extra powdered ice tea would dissolve in the water. There is a fixed ratio of how much a substance can be dissolved at a certain temperature, in form of g powdered ice tea/g water. Therefore, if you increase the amount of water, the amount of ice tea able to be dissolved will also increase by that same multiplier. To go back to the auditorium example, adding more water would be equivalent to adding more chairs for people to sit in.

Another method, while unrelated, can also help. If you increase the temperature, then you could also dissolve the extra powdered ice tea. One of the factors that affect solubility is temperature, which dictates that the solubility of most solids (powdered ice tea) and liquids (water) are directly proportional to temperature. Thus, if you increase the temperature, then more can be dissolved.