Why is it that drinking caffeinated tea does not help one to stay awake?
It could be tolerance if other caffeinated beverages are usually consumed, however it also depends on the type of tea itself, and other properties, such as how long the tea has steeped for.

For example, brewed black tea can have anywhere from 14-61 mg of caffeine, which is a large range and depends on which tea you are drinking specifically.  Whereas if you were drinking green tea that has not been steeped for long, it can have 24-40 mg of caffeine, which at the lower range will probably not help you stay awake.

From MayoClinic.Org:


  The actual caffeine content of the same coffee drink can vary from day to day — even at the same coffee shop — because of various factors, such as roasting and grinding, as well as brewing time. The caffeine content of tea also is affected by how long it's brewed. 


Caffeine beverage levels can be found here:

http://www.mayoclinic.org/healthy-living/nutrition-and-healthy-eating/in-depth/caffeine/art-20049372