Diagnosing Unexpected Errors

Simple question.  When a web service (or I suppose anything else for that matter) returns an "Unexpected Error" and provides an Error ID, is there any way to get more information about what's wrong?  My experience with web services has been very frustrating because anything but the most trivial mistakes are punished with these useless responses, and I end up spending most of my time on trial/error debugging.

I apologize if this question has been answered already, but as you might imagine, searching the forums for "unexpected error" produces an epic list of threads.  Reviewing a few samples suggests that further detail exists, but that the only way to find out the nature of the failure is to file a ticket with support, which as a rather serious problem for a "self-service" customer.  I would expect paid support customers to be annoyed as well, given my experience with the availability and responsiveness of support.