Is it normal to find something in tea?
Short Answer: No. Change your tea vendor.

Long Answer: 

My sweetie is a tea drinker, and gets Chinese teas from many premium vendors in the US, including Ten Ren, Jasmine Pearl, Silk Road, Vital Leaf, Red Robe, and others.  We currently have a kitchen cabinet so full of teas that bags fall out every time I open it.  Most of these teas are from China, as that country is both the origin of tea and the biggest producer.

In 10 years, I don't recall ever encountering garbage like you describe in tea.

I suspect that you are getting your tea from a bargain-basement source, were you buy tea for $10/lbs or less.  That's going to be one of the things that makes cheap vendors and grades different from premium ones.