Thought I'd add my two cents here - It's not quite about flying the real thing, or a 737, but I think it's as close as you can get without an ATP.
I have about a hundred hours of real time in a 172, but most of my time is in sims, and usually smaller planes; I recently managed to book a couple of hours in a full motion Level-D 767-300 sim, so in preparation, purchased a copy of Level-D's 767 sim and managed to get it running in P3D; the graphics are a little outdated but for the most part, all of the components are there - enough to familiarize myself the with flight deck and handling characteristics.
I spend a few hours doing circuits, ILS approaches and landings in various wind and visibility conditions (nothing extreme) just to get the procedures down and to get a handle on things like timing and power settings, etc.
During the briefing for the Level-D flight, the instructor indicated that I would probably crash, and not to be surprised if he had to freeze the sim at any point (full motion sims get get pretty rough!)
My copilot for this flight (we switched back and forth on roles) had no real flying experience, but a fair bit of flight sim experience; I thought I'd have an advantage having some real flight time, but it turns out it only made me more aware of hitting and holding my altitudes.
When we actually did the flights, both of us managed to do the take-off, climb to 3000 in the circuit, intercept the ILS and land without "breaking" the plane; the landings would have been a little hard and passengers would have commented (we assumed we were flying a cargo plane) but they were within limits so nothing would have busted. Some of the flights were even in mild turbulence and with reduced visibility.
As for taxiing, in a full motion sim, you get a really good feel for how much you're braking and turning, so it actually isn't that hard - turning is easy since you use a tiller - you just have to remember that you're sitting way in front of the front wheel and you're in a long plane when you make the turns (stay off the grass!) Also, the instructor warned us to be easy on the brakes - they are sensitive, but still require quite a bit of force.
So I think is is possible to go from sim experience to fly the real thing, but of course, there's lots of conditions attached; in our case, there was no cross-wind and we flew at dusk so the runway lights were easy to spot - this meant that we could switch to visual and use the PAPI lighting as a guide; we didn't have to worry about most of the systems - we started with the plane powered up and ready to go, including having the ILS tuned in; we hand flew the plane, so we didn't worry about the FMC or autopilot, just the basics including flaps and gear; the instructor gave a few pointers along the way - something someone could do over the radio - but never had to intervene or pause the sim.
It's actually quite amazing how close some of these sims get to the real thing; yes, a real plane (and a level-d sim) feel different when you're at the controls - the yoke is much heavier and has a much greater range of motion, the "monitor" is a LOT bigger and there's a lot more to take in, but the response to control inputs, power settings, etc. are all very close.
Oh, and if anyone wants to know, was it worth the big bucks? Every penny!