Correct. In 1957 or 58 (I can't remember which), the Fed increased interest rates to remove excess out of the economy. Economy went into a fairly mild recession. The Fed gets blamed for causing the recession (the Fed Chair at the time said the Fed should to take away the punch bowl...that stopped happening). Nixon loses to JFK (remember he was Eisenhower's VP, so Nixon blamed the Fed when he lost). And the cycle that led to the inflation of the 70s (where monetary and fiscal policy is timed to the election cycle) begins, tacit political involvement).
Also, before 1951 (and for a period of years after, although the formal break was 1951), the Fed wasn't functionally independent from govt. Because the war debt was so large, the Treasury used the Fed to press interest rates down so the debt could be paid down (it continued after 1951 because the debt was still really huge, note the similarity with your hypothetical). So the period at the end of the 50s was the first real test of Fed independence.
Again, I don't think people today understand that Fed independence is clear legally but has been more flexible in practice. Why? Because setting interest rates is inherently political. And there is an asymmetry: the incentive is always to be loose. The lesson is that there is no real way to get around political control, because the temptation is too great. I also think that policymakers should rely more heavily on macroprudential policy to take the heat out of markets (this is happening in the UK) because normal monetary policy is so asymmetric.