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?

Every solute has a certain saturation level for a given solvent, at a certain temperature. This is the maximum amount of solute that can be dissolved in a given quantity of solvent at a certain temperature. For example, say the saturation level of sugar is 20 g per lt of water at 20 degrees Celsius, then only 20 g of sugar will dissolve in 1 lt of water at that temperature and any extra will...

Every solute has a certain saturation level for a given solvent, at a certain temperature. This is the maximum amount of solute that can be dissolved in a given quantity of solvent at a certain temperature. For example, say the saturation level of sugar is 20 g per lt of water at 20 degrees Celsius, then only 20 g of sugar will dissolve in 1 lt of water at that temperature and any extra will simply stay undissolved. The remaining quantity of solute can be dissolved by one of the two (or both) methods: increasing the solvent volume and/or increasing the temperature. Adding extra solvent decreases the solute concentration below the saturation level and allows more solute to be dissolved. Increase in temperature of solvent increases the rate of dissolution and hence more solute gets dissolved. 

Note that there are a variety of powdered iced tea products available in market, some of them dissolve in cold water, while the others dissolve in hot water. If the powdered tea mix, we are discussing here, dissolves only in hot water (which means solubility is very low in cold water) then addition of extra cold water will not help. And temperature increase may be the only option.

Hope this helps.