Most incredible engineering feat in this movie was anchoring the winch on top a pallet and pulling 10x its weight without moving. Love the movie though.

Fitzcarraldo
http://www.imdb.com/board/10083946/?ref_=nv_sr_1

In
The African Queen
Charlie Alhnut had to build a forge in order to replace the broken propeller so they could continue their journey down the river. Later on in the film he designed and built a couple of torpedoes so they could ram them into the Louisa and sink her.

The latest entry in this category is "The Martian."

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Five Came Back -
Plane crashes in the jungle, and the survivors have to get it fixed and clear a runway, before they can leave. Can't say too much more than this without resorting to a ton of spoiler tags, but it's a great movie. They did a remake some years later, which was good, but not as good as the original.
How sad, that you were not born in my time, nor I, in yours.

There was a quite good, "made-for-tv" movie, "The Flight of the Birdmen", early 1970's. I've never seen it again, but when I looked for mention of it on imdb.com, others remembered it fondly as well. I believe that it was directly inspired by "Flight of the Phoenix", which would have come out 6-7 years early. I wish the movie would be released on DVD!

The Wages of Fear came to mind too, dealing with overcoming obstacles. The Great Escape fits, and A Bug's Life same idea of control overcoming chaos.
Bad outcomes can be seen in some of Crichtons novels like The Andromeda Strain, The Great Train Robbery (clockwork precision) and Jurassic Park of course. Park is a classic for showing improv's ability to deal with faults & breakdowns. Airframe 1996 is also a very gripping book about the complexity of technology and overreliance on it.
Phoenix is a terrific story especially since uses the human element so well compared to most recent techno thrillers. The latter flatten the characters to play up the FX razz. Film people like that since means less set time, collect pay, and on to next quickie.
Just saw Abandon Ship 1957 and it's haunting me. Set on the open sea (like the desert in that way) and who is critical to survival & who's not & who decides.
For now I'm leaving the life and death genre aside for a while!

Several flicks spring to mind.
Land of the Pharaohs is a movie that Howard Hawks did to show large numbers of men at work - the original idea was about building an airstrip in China during WWII. The final project - set in Ancient Egypt - is a pleasant movie about building pyramids and the traps inside them. Closing sequence is impressive.
The Wages of Fear (Le Salaire de la Peur) - Not a lot of pure engineering, but plenty of technical problem-solving, as four men try to get two trucks of nitro through the mountains of S. America.
10 Commandments - Not typically seen as an engineering movie, but the scenes of Moses building the city feature some solid scenes of old-school engineeering, particularly the obelisk scene.
The Great Escape - some engineering, but more at a systems level - putting together all the parts of the escape, and dealing with managing the people who have to solve the problems.
Most caper films share this element with The Great Escape.
A Bug's Life deals with a geek trying to solve a problem and pitch it to management, The sequence in which the plan is pitched, in succession, to the initial team, the leadership of the ant colony, and the the populace thereof, is a particularly well-turned sequence.
Hope this helps.

It's not seen too much, but "No Highway in the Sky" is a great aeronautical engineering movie. James Stewart (again) as an aircraft designer who finds a fault in the new "Reindeer" design. The special effects are hokey, but the story basically foretold what happened with the DeHaviland Comet a couple years after the movie came out. (Sort of like "The China Syndrome" coming out just weeks before the Three Mile Island nuclear accident.)

Well, how about The Dam Busters?
This is a true story of how the RAF was able to bomb the dams of the Ruhr valley, that would flood one of Germany's main industrial regions. They used a special bomb that would skip across the water. Though some things were changed to keep certain secrets secret, this was a rather ingenious, if rather unusual, means of precision aerial bombing for a day and age when "precise" meant the bomb landed withing 200 yards of the target.

In ICE COLD IN ALEX the characters are trying to climb a huge sand dune to get their truck to Alexandria. They can't make it in forward, so they back up inch by inch using a winch type device. It's the climax of the movie and well done.

Just finished watching Phoenix, am about to put in Murphy's War, with Peter O'Toole. Perfect example.

Not exactly the same sort of movie since it's a comedy, but I really enjoy Those Magnificent Men in Their Flying Machines. Watch the special features on the DVD. They actually reconstructed many historic airplanes, tho making them safer than the originals. I always thought it would be fun to build things like those.

Would that be Last Flight of Noah's Ark?
http://www.imdb.com/board/10081031/
A B-29 bomber is converted into a saling boat, very well realised.

i seem to remember a movie where a flying boat is turned into a sailboat.
can't remember much more than the image of the bird with a mast and sail.
maybe someone else can?
I did not save the boy, God did. I only CARRIED him.

Isn't there some film about people trying to lug a Spanish Galleon over a mountain range for some reason? A mammoth undertaking, even with the ship dissasembled.
it was Fitzcarraldo . great movie, true story!

Isn't there some film about people trying to lug a Spanish Galleon over a mountain range for some reason? A mammoth undertaking, even with the ship dissasembled.
More loosely there's Sahara, involving a search for an ironclad battleship in the desert. There are a few examples of ingenious improvisation in that one.

"First Men In the Moon" (lunar landing in the mid 19th century)

I asked myself the same question, and here's the best I could come up with in the way of epic stories of engineering and human resourcefulness- Dambusters, Das Boot, Bridge on the River Kwai, Road Warrior (seriously!)
In all these stories characters are forced into desperate situations in which they must improvise with the scant materials they have to create something original and brilliant. In all these scenarios, the solution is ingenious, but at the same time has a childlike simplicity to it, just like Dorfmann's basic premise of rearranging the limbs of the aircraft, like a giant lego model. In all this scenarios the solution is also visually spectacular, which is why the stories lend themselves so well to film treatment.
However, Flight of the Phoenix is and always will be the greatest story of this kind for me. I've just watched it for the first time in about 15 years, and it had me in tears.

Apollo 13 comes to mind, though I can't remember how much of the actual details about the engineering that they go into. Mostly, I remember the scene where the NASA employee shows the team of engineers all the components that they stranded astronauts would have to work with, and for them to create a solution with that.