Not quite right, Josh, but it takes a while to grasp and it's not entirely intuitive.
You don't go any deeper with a 10 minute exposure than you do with a stack of 10 one minute exposures. The difference is that you will incur read noise once in the first example and ten times in the second. How big a difference this makes will depend on the characteristics of your camera.
I'd recommend reading a decent reference on the topic. Craig Stark has written some good articles. See the Signal to Noise series here:
http://www.stark-labs.com/craig/articles/articles.html
The book
The Handbook of Astronomical Image Processing is a very interesting and informative read too.
Cheers,
Rick.