AA are very understanding. I missed a flight DFW - PSP for a wedding due to my own incompetence (I actually sat at the gate and watched my plane depart - I had it in my mind that the flight was later than it actually was, probably because I was full of cold and dosed up with various medications). AA did their best to find me another flight that day but couldn't, and so I was booked on the first flight the following day and given a "Distressed Passenger" rate coupon for a local hotel.

AA also has a published policy regarding misconnects between AA and other OW carriers (and vice versa) that includes flights on separate PNRs. That said, I have never had to call upon that policy and I tend to err on the side of caution when flying on separate tickets. I had two separate bookings for this Saturday coming (one LAS - DFW - IAH on AA, and then - three hours later - IAH - LHR - NCL as an Avios redemption with BA) but got cold feet last week and moved the first set of flights to the day before, Friday.

I'd rather have a relaxing evening in Houston safe in the knowledge I was in situ for the following day's flight back to the UK than a sleepless night in Las Vegas followed by a 0330 alarm clock for my early flight to DFW, and then fretting about our connection on to Houston.

If you haven't been to Chicago then it's definitely a city worth visiting. You can take the blue line from the airport to downtown and have a pleasant evening wandering around and taking in some of the sights.