One thing that I often relate to students is to learn to live with unusual differences in hands if there is just no way to overcome it.
With fat fingers for instance, you might be attempting to play a chord exactly as shown in a book or video. However, just because you might be muting a string or not able to play the chord exactly doesn't mean all is lost.

A lot of times the guitar is just a backup instrument for a vocalist or band, so there isn't the need to always play all the strings of a particular chord.
Even if you can manage just 3 notes to play, you've essentially covered the basics of a chord.

If all else fails, you can always turn to alternative tuning. I even use this myself when recording because I can get a smoother sound and hold down chords for longer periods of time.
For instance if I'm playing a song with all major keys, I'll just tune my guitar to open G. Or use Drop D tuning. Sometimes I'll put masking tape over a couple of strings if they're getting in the way.

When I first started playing guitar I had a similar problem.... not fat fingers... just too dumb to figure out how to play chords.
So I just tuned my guitar to an open G chord and used my thumb to hold chords down while holding the guitar in my lap.
I played guitar like that for a couple of years and even though I was limited to certain songs.. it still allowed me to play, as well as develop a lot of right hand strumming techniques that I would probably have missed if I'd spent all my time concentrating on my left hand.

There are all kinds of alternate tunings you can use... some of them are listed here:

http://guitartricks.com/lesson.php?input=1754

You'll often see girls using alternate tunings because they have a similar problem in just not having enough finger strength. With an alternate tuning, you have to experiment quite a bit but it'll come to you eventually.

Anyways... just try to do your best. Don't always feel that you absolutely have to play all the strings in a chord. If you find a finger is muting another string accidentally, over time you'll just get used to playing it that way and your mind will subconsciously adjust your playing style to suit it.
Just remember that even though there are suggested 'proper' ways to play, anything goes as long as it sounds good.

As an example, I remember being told very early on to never use my thumb. However, being the rebel kind of guy, I used it just to spite the guy. I learned over the years that there are several chords that can't be played without using the thumb so the lesson I learned from that was to not always take all advice as 'definitively' the correct answer... and I tell my students now... if you want to use your thumb go ahead.. heck, if you want to use your left elbow and your nose it's ok with me as long as it sounds good.

Another point I'll make, and something that still affects me to this day is that very early on I noticed that I couldn't hold my left hand fingers close to the fretboard at all times. No matter what I do, even now, if I press down a note with one of my first three fingers, my pinky finger shoots straight out.
Nothing I do will stop it from doing that because it's just the way my hand is built.
So my first hurdle to overcome was trying to play fast leads with a little finger that was constantly jumping 2" off the fretboard all the time. It majorly slowed me down and to this day, it still does.
However, rather than letting it limit my playing I learned to use an alternative way of playing leads by tapping them with two hands. After many years of practice I was able to play leads faster using this method than anybody else I knew. I distinctly remember at the time that I was learning to play like this, I always felt like I was cheating or doing something wrong.
But now after so many years I can look back now and think.. 'ya know, if it wasn't for that little finger problem, i'd never be able to play half the stuff I can do now so it was actually a blessing in disguise.'