There is one great lie in the digital age, that you can fix anything in post-production.
Perhaps that is an exaggeration, a competent digital artist can take the canvas of a photograph and use his skills to recreate an image (or at least construct it as it should have been). However, this takes time and great expense and at that point you start to wonder why you even bothered to start with the base of a photograph.
In the above example, taken more than a year ago, I was just toying with manual exposure and managed to hit just the wrong combination of aperture, shutter and ISO as I moved from outside to in to get a very dark photo of a very attractive marine biologist. Bringing up the exposure the result is noisy and full of purple fringing in rather annoying places, and the focus makes it look a little more like a painting than a photo.
Individually most of these are not insurmountable challenges. Color and exposure are especially easy to compensate for (unless either is completely shot) through conversion to black and white. Making an image monochromatic reduces the amount of information in an image, which in turn makes it a lot easier to work with from a recovery perspective.
It also has the advantage of looking dramatic and artistic, done right there's something very special about black and white photography.
That isn't to say every black and white image you see is a cover up of a photographer's mistakes, just as often it's a purposeful choice to heighten the impact of a photo rather than bring one back from the dead.
Poor composition can be fixed in post-production as well. If a background is cluttered or there is too much negative space it's a breeze to crop the photo down to eliminate unwanted objects. If the framing is too close to crop a photographer can apply a blurring action to the background that makes all of that clutter run into itself and seemingly disappear while the subject pops out of the photo.
The real challenge is bring out information that is no longer there for one reason or another. In any number of procedural cop shows or science-fiction movies they're enhancing grainy digital photos to recreate faces from a few pixels in a crowd or spotting some other minute detail within mere moments. In truth digital photography is a binary world, either you've got the information there or you don't.
A common problem with digital photography are blown highlights or shadows. Here the camera cannot capture the full range of a photograph (say bright sunlight versus deep shadows) so it turns dark areas into pools of black and bright into puddles of bright white. The information in those places, shapes in the shadows or the outlines of clouds is gone from the photograph forever.
Another issue is the crop. While above I said it's possible to use cropping to fix sins of composition it's only possibly if your first image had a good deal of space. If the opposite is true you won't be able to crop or recompose the photo, which might make it tough to print. At the other extreme if your subject disappears into the photo when you recrop you may find that the subject is not in focus, or there is not a favorable amount of detail.
Focus and motion blur are also potential deal-breakers especially as you reach extremes. At the moment there aren't any good ways to easily bring a subject into focus.
The biggest problem, especially in wedding photography, isn't about the photograph that didn't turn out quite right, it's the photo you didn't get at all. In those important moments, a first glance or first dance if you don't get it right the first time then that moment is gone forever and no amount of work in the digital darkroom will find it.