In deciding to insure a piece I look at whether, if I lost or damaged that piece, I would:
1. Want to replace it; and
2. Be able to easily replace it (and/or want to) from my own funds (self-insuring).
If the answer to 1 is Yes, and 2 is No, then I insure.
If the answer to 1 and 2 is both Yes, then I don't insure.
If the answer to 1 is No, then I don't insure.
My antique ring in my avatar is insured. As is my art deco dinner ring. My answer for both of those items in my little questionnaire above was Yes to 1, and No to 2. I know neither can be replaced exactly in the same sense as both are antiques, but I would rather have insurance to help me replace them with something than just wave goodbye to them and what value *I* ascribe to them (whether it is what I paid for them, or how I enjoy wearing them day to day, etc). I have these rings specifically insured under Jeweler's Mutual.
For items that I could replace with my own funds, or would not want or care to replace if something happened to them, I don't separately insure (depending on the loss/damage they may fall under my homeowners, or they may not, but I am not too worried about it).
Since you raised the sentimental aspect...as an example, I have some very sentimentally valuable, but not very monetarily valuable pieces (mostly just heirloom pieces) that I don't separately insure as while I could afford to replace the item with something similar if I hunted around on antique/vintage sites I could not replace the sentimental aspect which is what is important to me. So for those, my answer to my questionnaire was a mixed Yes/No to 1, as I would ideally want to replace them, but I practically couldn't really replace them in a way that made them "valuable" to me again, if that makes sense?
Now and then I pick up little baubles on LT or DB or whatever that I also just don't bother to insure. They are more just for fun and if something happened to them, I would either not bother replacing it, or would just get something else entirely. I am just not that attached to them.
I HAVE made a loss claim for a ring before, and I was very glad to have insurance (via Jeweler's Mutual) when it happened. It is personal preference, but I am someone who regularly insures against various risks in my life if to it makes sense to do so, and so to do so for certain jewelery is no different.