What you are most likely dealing with is multiple anchors that are so close to each other that they appear as one point. When a stroke is applied it reveals these imperfections.
Using the direct select arrow (shortcut - "a") click the anchor on one of your "round corners." Now drag that anchor, if there are multiple anchors in the same area you'll know right away because you'll see a new corner being pulled from the lump of points instead of displacing the arcing part of the C shape. If this is the case, then continue. (Hit "undo")
Now you'll want to get rid of these extra points using the pen-minus tool but this can be a real pain. What will most likely work just as well is, use the direct select tool, drag a selection box around one of the corners (effectively selecting all of the lumped up anchors), right click, choose "average", select "both" and hit ok. Now all of the anchors are exactly on top of each other and AI will treat them more like one anchor.