This happened to me and it drove me crazy. Those that have posted are absolutely correct: the clips must engage the groove in the spindle or the handle will fall off eventually. In my case one of my window crank handles had the teeth stripped so I just assumed it was not engaged correctly and thus got stripped. I got a generic handle and put that on until I could get to my Chevy supplier. The generic handle fit fine but looked a bit off (clips engaged). I got the new correct Chevy handle and tried to put it on. The clip would not engage totally with the “correct” handle. This annoyed me greatly so I pulled another handle. The new handle would not engage on another window regulator spindle. So, I then drove the old girl to the supplier to figure out the mystery. We pulled all the handles (mix of old and new) and what we discovered was that in some cases the groves for the clips varied and you cannot get the handle on far enough to engage the clip; but only on some spindles, not all. What was even more perplexing was that the diameter of some of the spindles were just a tiny bit too big for all of the new handles (off shore repos btw). To the best of my knowledge all the regulator spindles were original.
I told him I would find a way and left with a new handle that at least seemed to have the groves correct. I reamed out the new handle with a drill bit on a manual brace. It took off a minute amount of material so it was the exact same size as the regulator spindle (7/16 if I remember right). It barely touched the teeth.
So, what I think is that the sizes of the old spindles do not always line up with the new repos; but sometimes they do. And the clip groves are too high in some cases. One has the peer down between the handle and the door panel to see if it is truly engaged.
Ah, but the mystery of the old cars is half their charm.