The “false positives” noted by the NYT article refer to cases where the viral nucleic acid was detected, but at “low” levels.
There are many reasons why a sample my have a low target amount:
1. There is high variability between samples taken, even severely ill patents have days with no/low detectable target followed and preceded by high viral load measurements.
2. Patient is at early stage of disease, some of these progress to very high viral load, some remain very low and are typically asymptomatic.
3. Patient was exposed to virus/nucleic acids, but never truly infected. (it’s estimated that it takes ~1000 infectious viral particles to cause an infection in ~50% of people)
4. True false positive- patient had no target nucleic acid, some contamination from collection through processing introduced the target. This is typically <1%(It is extremely rare for the vast majority of these tests to have false positives due to chemistry or off-target amplification)
Initially, the focus was on getting the best sensitivity possible, as the risk of a false negative spreading to others far outweighed the risk of a false positive self-isolating. Now that a single false positive can shut down a classroom/business/etc. more focus is being paid to determining if someone is truly a risk to spread to others.
The NYC article extremely overstated the utility of Ct values by themselves. While Ct values can give some semi-quantitative value, the variability between assay, specimen type, lot, etc. makes it impossible to draw any conclusions from that single number. Any test that wanted to look at viral load would need to run standards to give a quantifiable number, and the value would need to be based on copies per number of cells collected (for most sample types) to have any ability to compare.
Here is a good explanation: https://doi.org/10.1016/S1473-3099(20)30424-2
tl;dr: The vast majority of positives came from patients with CoV-2 nucleic acid in their sample. Any effort to further stratify risk by viral load will take much more than the information we have on hand.