Nope. Sometimes the string will sneak up inside you and you have explore a little bit to find it in order to pull the tampon out but the tampon can't go anywhere but your vagina. The opening of the cervix at the bottom of the uterus is too small for so something as big as a tampon to get through. Once, when I couldn't find the string on my tampon, I panicked and wondered the same thing! I had to put my fingers inside my vagina to search for the string, but of course it was there and I pulled the tampon out. Since then, I've gotten more comfortable with putting tampons in and finding the string if it sneaks away, but man that first time was something else!
Good news, even if the string disappears you can't lose a tampon inside of you. Full disclosure: I once was so tired when changing a tampon that I put a new one in without taking out the old one. Once I realized it, it did take some work to get the first one out, but your vagina is not as deep as you might think. I actually used Kegel exercises (squeezing the vaginal muscles) to push it out. If you just can't get a tampon out, don't be too embarrassed to ask for help. You will not be the first woman this has happened to, and your doctor can help get it out - just don't leave it in there!
I had this same fear when I was first considering using tampons and I am happy to say the answer is definitely, no! Although it is possible to get the string caught up in there and forgotten. Been there, done that! So always be sure to remove the last tampon inserted when you are changing your tampon or at the end of your period. It sounds like a `duh' kind of thing, but I'm sure I'm not the only one who's done it before.