Don't really have too much previous exposure to the staggering feature you are referencing, which I am actually finding in the Profile Labels dialog box as opposed to the Profile Properties. However, from what I am seeing the labels actually have to overlap for the Staggering feature to be implemented by the Civil 3D, thereby resolving the overlap.
So unless your Major Stations and Text Size are such that the Major Station Labels overlap you would not see the affect of Staggering. To understand this better I created Minor Stations labels in addition to the Major Stations labels, and set Minor Station Increment to 5 and turned on Staggering, in order to make sure I got overlapping labels.
If your labels are overlapping, but the system is not implementing the staggering, not sure what is going on there. Maybe something to do with the Label Style, but that is just a guess without being able to look at the file.