I can’t visualize what you are describing. Are these a new set of points?  I do know there has been quality issues with some Asian made points sold as replacement parts for our airheads. EME sells points by NORIS, which still come from Germany, I believe. It might be that you need a better set of points.  Nothing should be popping apart while you trying to adjust the point gap.

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I have been thinking about this thread. David is right about sub-standard parts (China) in the supply chain, but that was a few years back. These parts could still float up in the stock of parts available from some suppliers.  In reference to David's picture, I am assuming that there is a dog of some kind on the back side where the point set is "hinged" for the lack of a better term. That would be the pivot point for making the necessary adjustment to the point gap. Now then, if the points base plate or the carrier plate where the points attach is distorted in any way, that could lead to the trouble you describe. 

The point gap can affect timing and dwell. It is important to get the gap pretty darn close to the specified clearance. 

Have ordered a set of Noris points.   We'll see how it goes.