In the past 40 years or so I've been riding Amtrak fairly regularly, I've only missed (or going to miss) connecting trains 5-6 times.
In one case 30+ years ago, the Broadway Ltd was cancelled due to blizzard conditions and they booked me on the Lakeshore Ltd to NYP to connect to a different train to Florida the following day. They also paid for a hotel in NY as the Lakeshore arrived about 12 hrs late due to the blizzard, but that was figured in so I made the rebooked train with roomette to Florida that trip.
I've been bused to make connections 3 times that I can recall. Not a terrible experience, other than forgetting my hat on the bus at Indianapolis to 'meet' my roomette at midnight a few years ago.
It was 3 or 4 years ago that I was on a 4-5 hours late #8 to Chicago and was booked out on #30. They boarded an Amtrak 'agent' at La Crosse (or was it Wisconsin Dells?) and everyone with connections in Chicago were instructed to meet with him in the lounge car. Mostly a waste of time, in my opinion. But he freely wrote out vouchers for hotel stays in Chicago and for transportation to/from the hotels for those who were willing to do that and take their chances on the next days' trains to <wherever> upon arriving Chicago. I've been through CUS too many times to count, and when we finally arrived about 20 minutes prior to the Lakeshore Ltd 9:30 departure, I literally RAN the length of the platform to the ticket counter (I NEVER check any baggage!), swapped my ticket for coach (it was an AGR redemption trip) and 'made due' riding coach to Springfield MA. I was 66 or 67 years old at the time, and riding coach for the night wasn't bad. I didn't get a seat-mate until Cleveland or Erie, so I enjoyed the seat all to myself that night.
I've learned from my own and others' experiences not to book same day connections in a handful of situations. Because of the sort-of-early departures of #50 and #30 from CHI, I won't book that connection coming from #8. BNSF does a LOT of trackwork every summer on their route STP-SEA/PDX so a 3-5 hours late arrival in Chicago happens too often during trackwork season (APR-NOV) in my opinion. Although 3 weeks ago, I was on #28 from PDX and came in only an hour late, which was expected per an email from Amtrak indicating trackwork in Wisconsin. Another 'don't do it' is #14 to #28 at PDX. 30 years ago when I was traveling 1-2 times per month on business, I made that connection many times. But in the past few years that Transitdocs.com and other sites show train performance, that connection fails too often. When I arrived PDX on 4/21 aboard #14, I would have made the connection to 28 without a problem. When I left PDX the next day on #28, #14 would miss making the connection by 2 hours. Another connection that I made a good number of times 30 years ago was #11 to #2 at LAX. I wouldn't try that one these days unless I really WANTED to be bused from <wherever> to connect. Going the opposite way, #1 to #14, I've done that maybe 6-7 times in the past 10 years or so and never missed the connection.
Perhaps the biggest difference between a missed connection on an airline vs Amtrak is the few accomodation types on a plane (a seat is a seat) vs seats/roomettes/bedrooms/handicap/family rooms on Amtrak. Airlines have multiple departures from A to B whereas Amtrak has only 1 per day, or 3 per week. So matching up accomodations is near impossible when a connection is missed, unless riding coach. A year ago, after a UP derailment damaged a bridge north of Sacramento caused the Coast Starlight to be cancelled, I had to change my plans in a hurry as I was booked on #14 less than a week later. I figured out a number of 'shorts' to stay in a sleeper LAX to CHI, but had to change rooms twice enroute to do it. It totally blew the Amtrak phone agent's mind when I had to explain the situation and that I already figured out the right combination of 'shorts', etc. She'd never had to deal with a crazy railfan before.
Through the years, how Amtrak handles missed connections has been tweaked and adjusted to do the best they can to get matching accomodations on the next train to your destination. But given that during the summer and holidays most sleeping cars are sold out, trying to get a sleeper for the next day is near impossible. HOWEVER...with the advent of cell phones, if you're on the train and know you're not going to make the connection, you'll be far ahead if you 'play with' the Amtrak web site to figure out the best solution for you HOURS before reaching Chicago. Maybe you can get a roomette or even a bedroom for half the trip, etc. Calling to change your reservations while you're still on the late train tends to baffle the agents. I did just that when CSX dumped a coal train on the B&O route out of WAS and I knew that #29 would not be running from WAS that evening. So I first had to convince her I knew that #30 was terminated at PGH and would originate there, and my connection in WAS would be blown, so I'll get off train #141 at NYP and take the Pennsylvanian to PGH to catch #29. She made the change and then I had to convince her to undo the AGR points penalty for changing my reservation! Fortunately, I succeeded.
So, in a jam, is riding coach for a night so awfully terrible for a usually in a roomette traveler? If it's not during peak travel times, it's OK. These days, I travel with a couple of inflatable pillows as I have to sleep slightly elevated. Using the same pillows to jam between the wall and the armrest works well in coach, too. Being prepared carrying some gaffers tape and piece of paper to block out a ceiling light in your eyes is always a good idea, too. Take some earplugs along too, as conductor(s), dining and lounge car crews use the PA way too often and after 10PM, usually stop, but at 6:30 or 7, your ears will be assulted!