It depends. I'm a mechanical engineering student and I have a lot of friends that are EE majors or EE grad students, and you literally can go anywhere, even more so than the ME grad student. ME students can go into any industry that involves moving parts (read: Automotive, Aero/Astronautics, HVAC systems, Manufacturing, Consumer goods, maybe even Biomedical if you have the right background)
EE students, however, as Eight Sage says, can do anything involving circuits, which, as you can probably tell, goes everywhere now; Software development, biomedical, VLSI (processors), communications (radio and phones), circuit analysis (don't even get me started) although a lot of my EE friends are doing software development and networking.
I'm doing ME because it seems a lot more interesting to me. I did a project a year ago as an ME, and when something fails (as it did quite often), we can point at the failure and say, "Oh, the problem's over here," and fix the problem accordingly. If a EE project has a component failure, you gotta break out the test probes and test each individual circuit. If there's a bug in code? You gotta go back to the code and debug, or worse, scrap the code entirely. I like being able to see my failures.