I've been adding new modules and blocks today. So many, I can't count them!
As I was testing out the Usage block newly installed, I decided I wanted to add this same block to the Front Page of Moodle. Duhh, there's no Block box appearing on the side panel at all.
What did I do? Thanks. Trix.
Hi Chad,
Nope, I'm the "only" system Admin. Once I deleted the Criteria Reference Block, it worked fine again.
I was going to comment on this in a separate post, but I perhaps it fits here? As I have been working through the blocks tutorial, I noticed, I think, the behaviour that is causing the confusion. It shows up quickly if you have limited the number of block types available in your installation.
Once you have added all the available block to a page, the Blocks dropdown list actually disappears. Perhaps to the developer, it made sense: "Based on your settings, you have added all the blocks you are allowed to add to this page, so hide the Blocks dropdown." However, to a new user, all you see is a user interface item I was using has now disappeared and I haven't asked it to! Where did it go??!! 
I will admit, it confused me for a little while. The only way I could get it to come back, as I think Ms. Sanders mentioned above, was to delete a block from the page, then the Blocks control reappeared.
Like Frank mentioned, perhaps this behaviour should be documented, (at least the Blocks tutorial should mention it), or better, don't hide the Block dropdown when editing is turned on, even if there are no more blocks to add. Just leave it up with "No more blocks" or some such item in it. I think that would lead to less confusion.
My two cents.
Carvell