Can a person become addicted to tea?
Yes, both tea (black tea at least, English speakers usually mean black tea when they say tea anyway) and coffee have caffeine which is addictive.

Apparently, being able to correctly reply to this question involves defining the word "addiction". My guess is that the negative connotation of the word "addiction" will cause some pressure to not use it.

Here's a recent review on caffeine addiction that says that caffeine is not addictive. I do not agree with their arguments but I'll leave it to the reader to judge.