Fountain of Knowledge

There's one problem with hitting both publishers and agents at the same time. If you land an agent, they won't know who you've already tried to contact. Even if you keep a list and give it to them, you may ruin your chance by presenting yourself not as well as an agent could. While I agree that you shouldn't limit yourself, you will want to be cautious doing both at the same time. My suggestion would be to query every single agent that possible fits your book and is reputable. If all those reject you and you still want to keep trying, then you can try going directly to the publisher.