CDC Suddenly Concerned About Covid 'False Positive' Tests?

The CDC has issued new guidelines for measuring "breakthrough" Covid cases (a vaccinated person contracting the virus), recommending that the PCR test cycles be dialed down to below 28 to avoid false positives. False positives could undermine support for the vaccine. This move makes sense...but what about an entire year of testing the general population with a cycle of 40 or above? Are some false positives more equal than others?