It sounds like you're talking about dependent typing, then, at least for integers? That's certainly a feature Rust lacks that seems like it would be nice, though I understand it's quite complicated to implement and would probably make Rust compile times much slower.
For ordinary integers, an arithmetic overflow is similar to an OOB array reference and should be trapped, though you might sometimes choose to disable the trap for better performance, similar to how you might disable an array subscript OOB check.
That's exactly what I described above. By default, trapping on overflow/underflow is enabled for debug builds and disabled for release builds. As I said, I think this is a sensible behavior. But in addition to per-operation explicit handling, you can explicitly turn global trapping behavior trapping on or off in your build profile, though.
In Ada? No dependent types, you just declare how to handle overflow, like declaring int16 vs int32 or similar. Dependent types means something entirely different and they are checked at compile time. SPARK uses something more like Hoare logic. Regular Ada uses runtime checks.
Whatever you want to call them, my point is that most languages, including Rust, don't have a way to define new integer types that are constrained by user-provided bounds.
Dependent types, as far as I'm aware, aren't defined in terms of "compile time" versus "run time"; they're just types that depend on a value. It seems to me that constraining an integer type to a specific range of values is a clear example of that, but I'm not a type theory expert.
Dependent types only make sense in the context of static typing, i.e. compile time. In a dependently typed language, if you have a term with type {1,2,3,4,5,6,7} and the program typechecks at compile time, you are guaranteed that there is no execution path through which that term takes on a value outside that set. You may need to supply a complicated proof to help the compiler.
In Ada you can define an integer type of range 1..7 and it is no big deal. There is no static guarantee like dependent types would give you. Instead, the runtime throws an exception if an out-of-range number gets sent there. It's simply a matter of the compiler generating extra code to do these checks.
There is a separate Ada-related tool called SPARK that can let you statically guarantee that the value stays in range. The verification method doesn't involve dependent types and you'd use the tool somewhat differently, but the end result is similar.
For what it's worth, Ada and Spark are listed separately in the Wiki article on dependent typing. Again, though, I'm not a language expert.
I'll look at the wiki article again but I can pretty much promise that Ada doesn't have dependent types. They are very much a bleeding edge language feature (Haskell will get them soon, so I will try using them then) and Ada is quite an old fashioned language, derived from Pascal. SPARK is basically an extra-safe subset of Ada with various features disabled, that is also designed to work with some verification tools to prove properties of programs. My understanding is that the proof methods don't involve dependent types, but maybe in some sense they do.
Dependent types require the type system to literally be Turing-complete, so you can have a type like "prime number" and prove number-theoretic properties of functions that operate on them. Apparently that is unintentionally possible to do with C++ template metaprogramming, so C++ is listed in the article, but actually trying to use C++ that way is totally insane and impractical.
I remember looking at the wiki article on dependent types a few years ago and finding it pretty bad. I've been wanting to read "The Little Typer" (thelittletyper.com) which is supposed to be a good intro. I've also played with Agda a little bit, but not used it for real.