Yes, I remember them and yes, we still have some. My DD was given a Princess Diana bear as well as a Shamrock bear as gifts and they are probably the most expensive ones we have (and as a result those 2 are kept safely away). The rest of them have been kids' toys only and most no longer have their TY tags.
I remember reading about Beanie Babies but they didn't reach this part of the world. I still don't have an idea what they look like and what the craze was all about.