So I got on the plane, and, unrelated to the story, met a fellow who introduced me to AWK and Hacker News. In the last 10 minutes, I took out my phone to see what gate my connection would be at, and saw that it would not be in Detroit, but in Chicago. The woman at the check-in desk had given me a ticket to the wrong flight.
I told a flight attendant, "I'm on the wrong flight," and she said "what do you mean you're on the wrong flight?" I explained the situation, and she told me to speak to the gate agent when I got off the plane, and that they would sort it out. I got off the plane and started explaining the situation to the gate agent, and as he was trying to figure it out, a flight attendant came off the plane and said to him, "you better be giving this guy free miles, and free upgrades, and free drinks." They put me on a connecting flight from Detroit to Wisconsin and I got in a few hours early. All in all, an interesting experience and a fun story. They didn't give me and free miles or drinks though.
To this day I have no idea how I got on that plane--whether it was a colossal blunder by the gate agent or, if, perhaps, by some coincidence, another person by my name (which is a fairly common name) was supposed to be traveling to Detroit from that airport that day and missed their flight.