Can you submit the same picture to more than one stock agency?  For example all of the big 4.

Yes, you can. To 20 or more if you like!  
But you have to realize that acceptance and rejections are different to the sites, so don't take that personal.
Strange enough the images that I have accepted by all the sites I uploaded them to, are mostly not my best images in my own opinion!  
Sometimes bestsellers on one site are not accepted at others or have no downloads there.
When you are going to upload to more sites you will propably discover that some sites will do better for you than others. Depending on your portfolio. So after a while it will be possible that you will decide to upload only to that stocksites.
To me Istock and Fotolia are not the easiest to get images accepted and Crestock is the worst. Fotolia does a lot of "overabundant" rejection reasons. At least to me!
Good luck!

PS: As reaction to alsfoodtown: You can upload single images exclusive to Fotolia or Dreamstime. That single images you can not upload to others of course...

We should write a "Microstock FAQ" to avoid repeated questions...