One bottle should be able to fit into the other with no difficulty. The inner bottle can be made of glass or plastic, but the outer bottle should be made of plastic. Make sure that the inner bottle also has a reusable cap.
The outer (larger) bottle needs to be cut apart, so that is why you should use a plastic bottle instead of a glass one. Glass is a better insulator than plastic under most conditions, so if you can find a glass bottle with a reusable cap, use that for the smaller inner bottle. The cap is important, though, so use a plastic bottle with a reusable cap if you cannot find a glass one that will work. A 1-qt (1-L) bottle and 2-qt (2-L) bottle will usually work for this project. If you are not satisfied with the size of these bottles, however, you can use two different sizes as long as the smaller bottle can fit inside the larger bottle with a little spare room along the sides.