Sorry if this is a dumb question but I've just watched a documentary on the cold war and given the sheer number of nuclear weapons produced by the US and the USSR and accounting that both valued them as "peace keepers", wouldn't it be possible that one or both of these nations tricked each other into believeing that they produced more nuclear missiles than they actually did?
Yes, it did happen, and we're aware of it. It tended to be caused more by faulty intelligence by the US, rather than false claims by the USSR, however.
There was a scare in the late 50s of a "missile gap" between the USSR and the US. This was caused by a mix of fears from the 1957 Sputnik launch and some vastly inflated numbers in the National Intelligence Estimate, which estimated the USSR could have 100-500 ICBMs when they in fact had no more than 25.
The issue was seized and brought into national prominence by JFK, who used it as a campaign issue to make his opponents look weak on defense.
At the same time, the US had secretly found that the missile gap didn't exist, thanks to flyovers of Soviet airspace with the U-2 that gave a better estimate of their nuclear capabilities. But they couldn't publicly release that information without jeopardizing the U-2 program. The gap was largely in the other direction, with the US having a much larger arsenal than the Russians.
In the end it had a negative effect. Khrushchev became convinced that Kennedy was dangerous and trying to overstate Russian nuclear capabilities as a justification for a first strike. This likely contributed to the decision to place missiles in Cuba.