Hi, Marie. Here are 2 tile questions and answers: Can tile be painted? Yes. Will the paint last? No.
One of the main advantages of tile is that it has a hard, smooth surface. This hard, smooth surface is more like glass since the glaze melts when it is in the kiln. Most other things that you paint have a porous surface with a microscopic roughness that tile does not have. This gives tile it's easy clean-up and stain resisting properties.
Some people will tell you that you can paint tile after you sand the surface. You can do this and the surface will will look great for about a day after it is painted. But eventually, someone will find the area that wasn't sanded quite as well and the paint will begin to flake off.
I guess that you could remove the old tile, have it reglazed and fired in a kiln and reinstalled but the cost would be much greater than just installing new tile.