北谷町 and 下町 The first one is pronounced Chatanchou and the second is Shitamachi. The second term is within an area of the the first one. Is a machi smaller than a chou? In the past was the term ノ・の used in between the kanji to indicate a machi, for example shita no machi?
In the east of Japan, Machi is used more often.
In the west, Cho is used more often.
Google “まち ちょう” and you will find a good picture.
If you study kanji more, you'll realize that both readings are acceptable depending on the circumstances. Learning what reading is correct in each situation is something that you will have to practice a lot before you get the feel for it.
In general, more populated areas tended to be read まち, and less populated areas tended to be read ちょう, but exceptions were very common. Additionally, as @Yosh as pointed out in the comments, size of the region doesn't necessarily affect the reading either.
For the purpose of addresses, in my experience a まちand a ちょうare mutually exclusive since they mean the same thing. So no, a まちcannot be in a ちょう, and the opposite is also true.