That's what happened. I was on the Sunset and we left 12.5 hours late from Los Angeles. Our conductor said our original trainset was sent as the eastbound Southwest Chief and we therefore had no train to board at 10:30 pm on Wednesday. Had to wait on the westbound Southwest Chief trainset to be serviced for our use as the Sunset. I opted to spend the night at the Days Inn across the street from the station as we were told the train would be ready around 6:45 the next morning. Of course it wasn't and we didn't leave until 11am Thursday.Great thing about the extremely late departure was that we saw some scenery normally passed in the night when the Sunset is on time, including the large sand dunes near Yuma, AZ and a rare daylight eastbound crossing of the Pecos High Bridge in Texas.
Before arriving into San Antonio at 8:15pm (by now almost 15 hours late) the conductor informed us that arrangments had been made for some of the Texas Eagle passengers to continue by bus north to destinations in Texas. However, checking around, I discovered that passengers traveling north of Texas were told they could stay on the train... the last two cars of the Sunset scheduled to connect with the Texas Eagle the following morning. Not wanting to be stuck on a bus in the middle of the night for 7 hours and arrive in Dallas at 4am, I decided I would just stay in my sleeper with the Arkansas, Missouri, and Illinois passengers. This option wasn't announced to the Texas passengers and many of them (including several elderly and disabled passengers for whom the bus ride would surely be uncomfortable) jumped on the bus like it was the best thing going and left around 9pm or so.
Still, we left San Antonio almost an hour and half late on Saturday morning and were making pretty good time until we got within about 3 miles of the Dallas station where we waited for more than 2 hours for a couple of freight trains to clear.
Arrived in Dallas last night, 27.5 hours late.
Nevertheless, an extremely interesting mix of fellow passengers in my sleeper car actually made this my most enjoyable Amtrak trip to date.