Well, not quite, but they did delay some flights.
On a flight last month from Amristar in India to London, passengers noticed a rat on the plane before it took off, and reported it to the crew. Passengers were taken off the flight, and eventually put on a replacement aircraft that took off 11 hours later. Officials were concerned the rat might chew through some of the plane's wiring--they said that the aircraft would have to go through a full suite of safety checks before it could fly again, and that would take a long time.
Harry Francis, Canadian Manager for Air India, said: "This isn't common, but it has happened before. It can happen on any airline, and it does." Very reassuring. Source
In 2003, a rat was seen on a plane about to depart from India for Riyadh. After fumigating the plane, the rat was found dead.
In 2009, there were six incidents involving rats seen on planes of Pakistan Airlines during flight. In one incident, a rat jumped on a woman, whose screams alarmed other passengers. source
In 2006 mechanics were removing seats on an American Airlines plane, when they discovered mice under the seats. source