After going through most of the answers here, I'd like to add a couple of thoughts.
Relying on XML Documentation Comments and expecting others to rely on is a poor choice. Most C# code I've come across does not document methods completely and consistently with XML Documentation Comments. And then there's the bigger issue that without checked exceptions in C#, how could you document all exceptions your method throws for the purpose of your API user to know how to handle them all individually? Remember, you only know about the ones you throw yourself with the throw keyword in your implementation. APIs you're using inside your method implementation might also throw exceptions that you don't know about because they might not be documented and you're not handling them in your implementation, so they'll blow up in face of the caller of your method. In other words, these XML documentation comments are no replacement for checked exceptions.
Andreas linked an interview with Anders Hejlsberg in the answers here on why the C# design team decided against checked exceptions. The ultimate response to the original question is hidden in that interview:
The programmers protect their code by writing try finally's everywhere, so they'll back out correctly if an exception occurs, but they're not actually interested in handling the exceptions.
In other words, nobody should be interested in what kind of exception can be expected for a particular API as you're always going to catch all of them everywhere. And if you want to really care about particular exceptions, how to handle them is up to you and not someone defining a method signature with something like the Java throws keyword, forcing particular exception handling on an API user.
--
Personally, I'm torn here. I agree with Anders that having checked exceptions doesn't solve the problem without adding new, different problems. Just like with the XML documentation comments, I rarely see C# code with everything wrapped in try finally blocks. It feels to me though this is indeed your only option and something that seems like a good practice.