Somewhere between 0 decibans and 50 decibans, depending on the question you are actually trying to ask. If you're asking "Is there some way to interpret QM equations without invoking the concept of atoms," I'd give that about 50/50, because I don't actually know QM and I could see that going either way. If your question is instead "How confident are you that you will continue to make observations consistent with atomic theory for the rest of your life," I'm quite confident that I won't see anything obviously falsifying the existence of, say, hydrogen.
For the "'H2O is a meaningful description of water" question, the same problem comes up: most of the scenarios in which it's not are scenarios in which this was a trick question, because this pattern-matches really strongly as a trick question. My estimate of the probability that I missed something rarely drops below 10% when communicating through asynchronous text with another human.
I'm not trying to deal with the complications of quantum mechanics; and I'm not trying to ask a trick question. (I'm just nowhere near as good at expressing myself as I wish I was.) At the end of your first paragraph, you come close to answering my question - all I'm hoping is to get more of a quantification of how confident you are
How much confidence do you place in the scientific theory that ordinary matter is made of discrete units, or 'atoms', as opposed to being infinitely divisible?
More than 50%? 90%? 99%? 99.9%? 99.99%? 99.999%? More? If so, how much more? (If describing your answer in percentages is cumbersome, then feel free to use the logarithmic scale of decibans, where 10 decibans corresponds to 90% confidence, 20 to 99%, 30 to 99.9%, etc.)
This question freely acknowledges that there are aspects of physics which the atomic theory does not directly cover, such as conditions of extremely high energy. This question is primarily concerned with that portion of physics in which the atomic theory makes testable predictions.
This question also freely acknowledges that its current phrasing and presentation may not be the best possible to elicit answers from the LessWrong community, and will be happy to accept suggestions for improvement.
Edit: By 'atomic theory', this question refers to the century-plus-old theory. A reasonably accurate rewording is: "Do you believe 'H2O' is a meaningful description of water?".