There have been a few American close calls. One of the most frightening happened in 1979 when a training scenario was mistakenly loaded into the wrong computer, leading to folks at NORAD believing that a full scale Soviet ICBM launch against the US had occurred. The error was determined when confirmation of the launches was sought via satellite and radar systems.

In 1960 a radar station in Greenland mistook the moonrise for a full scale Soviet attack. One reason for doubting the data was that the Soviet Premier (Khrushchev) was currently visiting New York. If the error had happened some other time who knows what might have happened.

In 1961 NORAD lost contact with several BMEWS (Ballistic Missile Early Warning System) sites. Since the communication lines with all those sites were supposed to be heavily redundant, this seemed like a very high confidence sign of a preemptive attack. Though it turned out the system was not quite as redundant as they thought it was.

In 1965 during the great Northeast Blackout several nuclear bomb detectors malfunctioned and indicated that cities had been destroyed by atomic weapons.

There have also been some other related incidents. For example, the most important bit of the "nuclear football" is just a 3x5 card with authentication codes printed on it. President Reagan preferred to carry that card in a pocket of his jacket (as did Carter). When Reagan was shot and taken to the hospital the card was lost track of when medical personnel cut off his clothes to treat his wounds, the card was later found tucked into one of his shoes left on the floor of the emergency room.

(There are several others as well, which I won't list here.)

It's hard to say exactly how close these close calls were without having greater knowledge of the precise nature of all of the nuclear command and control facilities, organizational structure, and processes, many of which remain classified to this day. My gut tells me that none of these were really all that close, as in general there were easy sanity checks that led people toward the correct conclusions that WW3 had not actually already started. However, one has to wonder. How would things have turned out if a different administration was in power, or if certain incidents had happened during more tense times, or if coincidental incidents had happened at the same time? For example, if the Northeast Blackout and erroneous signal of a nuclear attack had happened during the Cuban Missile Crisis instead of in 1965, would it have escalated?

(Edit: fixed a "brain typo" confusing Turkey and Greenland.)