It certainly would not be the first war, unwanted by both parties, started by miscommunication. It's easily done. Imagine you and someone else are pointing guns at each other - did the other person's buddy just say 'shoot' or 'shucks'? Are you going to wait to find out? And what if your buddy says it - you probably want your buddy to think hard about what they say and to speak very clearly and slowly.
That's why diplomacy is a profession which requires great skill and expertise, and why escalation is rarely used as a tactic by the professionals. Flying bombers which could contain nuclear weapons near the other side's cities, for example, might make jumpy people pull the trigger. Radars, nervous radar technicians, and their nervous superiors do make make mistakes.
A surprisingly large number of wars do start with errors. Off the top of my head:
The destruction of the ship the Maine by the Spanish started the Spanish-American War, even though the ship likely wasn't destroyed by the Spanish.
The Gulf of Tonkin incident, when the Vietnamese attacked U.S. Navy ships. started the Vietnam War, even though there was no attack (it was radar error).
Iraq's WMD program and alliance with Al Qaeda started the Iraq War, even though there was no program or alliance.