Spinning an object in a perfectly horizontal circle?

I was teaching my physics class the other day. We are doing centripetal motion right now. We just finished dynamics, and learning about forces, and now we just got into circular motion.

A lot of the questions in the textbook have to do horizontal circular motion, and a lot of the questions in the textbooks ask "determining the tension of the string when an object is spun horizontally". I never questioned these statements before, but then I started my usual dialogue with my students about all the forces acting on the object being spun... They started to list off: Tension, Force of gravity... oh oh, I thought. Why is the object not falling to the ground, if there is only one force acting in the vertical direction? Horizontally it is accelerated into the circle by the Tension force of the string, and that's why it is rotating. But what about vertically? I started to think about it the whole class. By the end of the class, I was convinced that there is no way that we can actually spin something perfectly horizontally - that it is impossible. But I still thought that maybe I was just not thinking of some other factor.

Later in the class, the students were doing an activity spinning objects: changing the radius, finding the speed of the objects. They then had to make a graph of v versus r, and analyze it. A simple activity, but I was really making sure I was paying attention whether any of the students could actually get their objects and string perfectly horizontally... and they couldn't. Further support for my hypothesis.

Right after class, I read all my textbooks on circular motion. None of them touched upon my dilemma. Then I asked Google, whether it was possible for an object to spin perfectly horizontally, the string and object in the same plane. No satisfactory answer, but I suspect I wasn't looking properly, since I can't be the only teacher asking themselves this seemingly simple question.

So here is my hypothesis: It is impossible to spin an object horizontally where both the object and the string lie in the same plane, no matter how fast you are spinning the object, because the object must have a vertical force opposing gravity. (I suspect this would come from the vertical component of the tension force of the string.)

Consequence: When asking questions about tension of the string spinning horizontally, one cannot use only centripetal force as the answer. This will give only the horizontal component of the tension. The vertical component of the tension must come from (opposing) gravitational force of the object... and then we must put these together.

Now that I thought about this over the weekend, I am so very certain that I'm right. But still, let me know if you think otherwise, or if you agree with me. I don't want to teach my students something completely bogus, and then realize that I missed something...

PLEASE HELP!!! (comment to help)