Before Hurricane Katrina, Amtrak’s Sunset Limited ran from Miami to Los Angeles. The trip took a little under three days.
Damage to tracks and bridges disrupted service between New Orleans and Miami. The damage has long since been repaired, but Amtrak has not seen fit to restore service (my private hunch is that they’re using the rolling stock made available by the shortened route elsewhere).
Here’s an archived New York Times article on the subject from 1981. Amtrak had been in service for a decade then.
Other than when the Sunset Limited route had been extended to include Jacksonville – Miami came later – there’s never been a direct coast-to-coast route in the US. Up north, however, Grand Trunk Pacific was running coast-to-coast service as early as 1906.
Part of the issue with going cross-country on one train, pre-Amtrak (i.e., pre-1971), is that the individual railroads ran their own passenger services, and there wasn’t (and still isn’t) one railroad which operated nationwide.
But, as most of the major railroads did service Chicago, that city became the focal point for transferring between an eastern train (which might have run on the New York Central / Pennsylvania / Penn Central line, or others) and a western train (which might have run on the Union Pacific, Santa Fe, or others).
In the late '40s and into the '50s, the Santa Fe and several eastern railroads offered run-through transcontinental service. Your trip would involve two railroads, but you would stay in the same sleeper car for the entire trip. The service was available from New York to LA (and vice versa, of course) via the New York Central or the Pennsylvania Railroad and the AT&SF. The service was also available from Washington to LA via the B & O and the AT&SF.
Also, the California Zephyr from its inception in 1949 until 1957 offered through service east of Chicago on the Pennsylvania Railroad in the single 10-R 6-DB sleeper the latter owned, Silver Rapids. The Zephyr’s service from Emeryville to Chicago was daily, but the eastern connection was tri-weekly due to the one car shuttling back and forth. Service was dropped due to lack of ridership. Cite.
I don’t know about southern Florida, but the Frisco ran the Kansas City-Florida Special with the Southern. One could transfer to/from Miami in Jacksonville. Major stops were Kansas City, Memphis, Birmingham, Atlanta, and Jacksonville.
The old joke was that “a pig can ride through Chicago without changing trains, but you can’t.”
As MonkeysBrother points out, in the postwar era a couple of railroads teamed up to offer through sleeping-car service. You could arrive in the morning on one train, remain in your Pullman room while the car spent the afternoon being shuttled through some South Side railyards, and then depart in the afternoon for the other coast. It’s doubtful that many people ever did this. People who regularly traveled coast-to-coast through Chicago instead would arrive in the morning, do some shopping at Marshall Field’s, have lunch at the Pump Room, see a matinee or maybe a newsreel, and arrive at the other train station for a 5 pm departure. The Parmelee Transfer Company meanwhile would have transferred their bags.
That’s still quite common in some European countries with a heavily centralised rail network, such as the UK or France. If you’re in France and want to travel from one provicial (i.e., anything outside Paris) city to another, in many cases you’ll have to change in Paris, but Paris has several terminus stations, serving different directions. So depending on where you’re coming from and where you’re going, you might end up having to transfer from one Paris station to another.
Compare that to the usual German approach of consolidating long-distance traffic, as much as possible, in one central station per city (“Hauptbahnhof”). Then again, Germany has a much more decentralised rail network that does not have one central capital hub that spokes lead to from all directions. My understanding is that this was also the idea behind the “union stations” in some American cities, which were jointly owned by several rail companies to consolidate their services for that city there.