As far as I know, the term "bounded array" is not really standard terminology for anything in particular. I would assume, however, that it refers to a static array, something like this: int arr;. It is still used today, for sure. When you just want a small set of objects or variables for the duration of a function, or as a data member of a class. However, technically-speaking, the C++11 standard introduced the std::array class template which can replace a static array and has a number of benefits, such as being copyable, and usable like an STL container.
The size of a container is fixed and cannot change.
Accessing indices outside of the container's range will produce an exception.
Can you clarify what you mean by "bounded"? I'm also not familiar with that as any kind of standard terminology. IIRC, the C++ standard only mentions "bounded" in reference to numeric ranges and syntax grammar; nothing about arrays.
In that example, the vals array would be called a static array or fixed-size array. But, the tests that are done in the get and put functions would be called "bound-checking" or bounded array access (I.e., the array access is "bounded"). So this might explain how you got the term "bounded array" from.