I must quibble a little bit. Most of the time, things like recovering the boosters and fairings were not stated to be impossible, just impractical. Not worth the effort. Not profitable. Those are the statements that SpaceX has shown were wrong. And that is an impressive accomplishment, no doubt. Yet they have this reputation for doing "the impossible" which is not quite accurate, unless you say that doing these things profitably is the impossible.
Yeah, I'll buy that.
Boeing is a private company. Lockheed Martin is a private company. Rockwell, Orbital Sciences, Grumman, Northrop, and other companies that were or are involved in building and launching rockets and their payloads are or were private companies. NASA designs and builds almost nothing; it creates plans and goals, contracts out the hard parts, and oversees the execution. This is not to knock NASA, it's the way government agencies to pretty much everything, and it has worked. A good deal of SpaceX's work has been done under NASA contracts, just like the others. SpaceX uses a different mind set which works in many (most?) ways better than their "predecessors" in the business, but to say it's because they're private is a part of the SpaceX mystique that's just not so.
And, we haaaave a winner! Yes, SpaceX is doing remarkable and exciting things. And WOW does Elon love to talk about it!
I was working for Orbital Sciences on development of its Commercial Resupply Service (CRS) vehicle, the space truck that brings material up to the ISS. SpaceX was developing the same thing, a very similar vehicle, under a virtually identical NASA contract. When the development, and later the missions, were mentioned in mass media it was always the SpaceX vehicle, with Orbital's as a footnote if mentioned at all. Why? Because Musk just loves the sound of his own voice.