By "double tapping", are you talking about your multiple start threads? Or are you talking about re tapping an existing hole? If the former, I have no idea why one lead would effect another lead. If the latter, then you are taking a spring pass and the expectation would be the thread would be larger than before, as with any spring pass with any other tool.
I don't know specifically about lathes, but in our mill, we rigid tap down to 0-80 in 304, 6061, plastics, etc. at 6,000 RPM (it doesn't actually reach that), and we've done many, many thousands of holes. We've never had an issue from the machine being inaccurate.