Should I assume my LMV324 is counterfeit?
If you tested the part correctly to see the output saturated at 1.3 to 1.5V below the positive supply rail then that certainly does not live up to the claim for an LMV324 opamp being a rail to rail output part.

It could very well be that the parts you purchased are cheap junk LM324 opamps that were remarked as LMV324 to make them appear to be better parts. If so I'd call that a counterfeit.