Oh yeah... sounds like tabascos alright. Those really do have tough little stems considering how thin they are. I would cut them; I use pruners for all my peppers, but I'm sure a sharp knife or a pair of decent scissors or shears would work as well. I remember last year before I got my pruners, I was drinking beer and wanted a tabasco pepper. I did the same you described after giving up on trying to bend the stem and "cut" it with my fingernails and ended up with the same outcome... tried pulling and had the end of the branch in my hand.
An easy, sharp, clean cut is almost always better than pulling, because there is less chance of damaging the plant. Not to mention, I've found that it gives you greater accuracy, allowing you to cut with less chance of accidentally hitting a new bud or flower and knocking it off while wiggling or struggling to get the pepper off the plant.