Well, at least you're not alone now. I've never seen or even heard of that issue before, but the description of it alone I find puzzling and the next one you run across I'd like to get a hold of here. So this doesn't really help much, as all I have on it are questions.
What version controller(s) have you seen this on?
Both axis of the Stick are affected? or only one? or all of them on the controller?
What are the voltage measurements on the Stick when it's acting that way? The Wiper (center lead) should measure ~0.9v and swing form 0v to 1.8v max with the shell removed. Actual is ~0.45 to 1.5v with the controller together. No two controllers will be spot on with each other also.
What makes it go back to working normal? or does it? and if it doesn't then how did the controller act before? That's an important one as testing before hand isn't always done.
What are you using to test the sensitivity? and have you tried another app or device to compare the sensitivity and confirm it's acting differently? Power cycled the controller, removed all power sources, batteries and USB cable and let sit for a bit.
The only way to increase/decrease the actual sensitivity of the Analog line is to change the rate the voltage changes, or change the FW in the MCU so it can then output a different value than what is input from the Stick. Changing the POT value doesn't do that, so 10k, 1k, 100k, it's all the same there as they are just used as voltage dividers.