What could cause individual false positives in ELISAs, while the rest of the plate looks fine?
Is your detection or secondary antibody aggregating? I've done extensive clinical ELISA development in the past and when this problem inevitably crops up for these assays the most likely culprits in order were:

Plate washer acting funny
Antibody aggregates binding nonspecifically to the plate
Inefficient blocking reagent/not enough time spent blocking
Person running the samples is mixing things up

Even if your false positives predate the platewasher malfunction, you could still have a leaky/clogged dispensing nozzle. Try rotating the plate 180 between wash steps or hand-washing and see if that helps.
For aggregated secondary/detection ab, give them a brief spin and pipette off the top, and for blocking increase volume/time.