I did have it tested and it is the type with asbestos. This is why I'm so concerned. It is not crumbling. You can walk on it and it feels very solid. Some of it did break when we took up the carpet that covered it---where the carpet was tacked down is where it cracked. We got suspicious after the fact and I had it tested. A good thick layer of regular old floor wax is a thought. I would have never thought of this.
I do want to say a big "Thank you" to everyone that is sending me advice. You've really helped calm my nerves a bit about this tile situation.
Thank you Mike. I am so confused. I really don't know what to do. Most web sites I go to say to not disturb the floor, just to cover it up. My plan was to cover it up with a layer of cement-leveler as an extra precaution and then put a new floor on top. What you said about the leveler making the situation worse puts me back to square one. It was on the back of my mind that a cement coat could actually soften or deteriorate the old tile or something and make matters worse.