I've seen it happen twice, once to me.

The first time was pre 2001. I was connecting in PIT for LGA on US. My incoming flight was late and it was the last flight out to LGA. I ran through the terminal and got there as boarding was about done. I boarded and my seat assignment in first was empty and I sat down. They then did a destination check, Burlington, VT. I got up to get up and they were coming to get me. They noticed the mistake in the boarding pass as they went through them before closing the door. There were two jetways in the same gate area and both were boarding. I took the closest to where I was running from and it was the wrong one. I made the LGA flight.

Second time I was doing a mileage run Vegas, also on US as I recall. This was after 2001. I was leaving Vegas for PHL in a 757. I get onboard in the last row of first and the guy across the aisle from me is excitedly talking on his cell phone. He is saying how great it is that they had put him first class and he had only bought a coach ticket. I didn'tay much attention then they did the destination check for PHL. He turns and says to me "did they just say we are going to PHL?" I confirmed that we were and he soda, "but I'm going to SFO". I assured him that he was going to SFO through PHL unless he got off before they closed the door, which was VERY soon. He bolted and found an FA and showed his boarding pass. They hold departure and get the gate agents, more FAs and they keep passing around his boarding pass, which as it turns out was an a UA flight to SFO. When we finally left they were still trying to figure out how he got on with a boarding pass from the wrong airline.