Trevor probably covered this, but I'll reiterate.
Keep in mind that a string works best when the ends are 'fixed': tied to something that is really massive and stiff so that they don't move. The problem is that if the bridge on the guitar doesn't move no energy will get out of the string to drive the top, and you won't hear it. You get a 'short' note when:
1) it's easy to move the bridge where the string is tied to it at that pitch, and
2) the guitar can extract the energy from that string quickly. These are related, of course, but not identical.
All of this is a function of the resonant structure of the guitar. Every worthwhile guitar has a number of 'resonances' at different pitches; structured ways of moving that are easy to drive and more or less well defined in pitch. Strings are the same way, as you can easily prove. Just touch a string at some point that divides it into equal parts and pluck it. If you touch it at the 12th fret you've divided it in half, and it vibrates an octave higher than it's fundamental pitch, which is twice the frequency. If you touch it at the 7th fret you've divided it into thirds, and get the interval of a 12th; an octave and a fifth, which is three times the fundamental pitch, and so on. Strings are pretty much one-dimensional structures, so the overtone series they produce is simple. Guitars are more complicated, so they vibrate in ways that are more complicated, and harder to predict as well. We often talk about these different ways of vibrating as 'modes', and sometimes refer to them as 'top', 'back' or 'air' modes, depending on what's moving the most. This is vastly oversimplified, but handy; none of these movements is 'just' in the top, or back, or air. The higher you go in pitch the more of them there are, and the more complicated things get.
The lowest pitched useful mode on most guitars is called the 'main air' mode, where the air flows in and out of the hole at a certain pitch. It's a 'Helmholtz mode', analogous to what you get when you blow across the mouth of a wine bottle. In this case, though, the top (and the back usually as well!) get into the act. Air moving in and out changes the pressure in the box, and pushes on the top, so that moves. It's easy for the string to get this going, and the guitar is very effective at turning it into sound, so it's a common 'short' note, often down around G on the low E string. There's another similar mode, often called the 'main top' mode, about an octave higher, where most of the action is in the top. It can be a problem as well, but not usually so much as the 'air' mode.
The next thing up in pitch is sometimes called a 'cross dipole' mode: the bridge is rocking sideways, with the treble side going 'down' while the bass side moves 'up'. Since the center of the bridge doesn't move the two middle strings can't push this one very easily, but the high and low Es can, and the A and B strings to some extent. This one often tends to come in at around B~247 Hz, right about the pitch of the open B string, so that can be a 'short' note too, but only on that string. The pitch is too low for the high E, and too high for a fundamental on the low E or A, although it might affect an overtone on those strings.
The next resonance up is often a 'long dipole' of the top, where the bridge is rocking forward and aft as the upper and lower parts of the top move out of phase with each other, similarly to the 'cross dipole'. This can be a strong resonance, and also an effective sound producer, but it's hard for the string to drive it, so it's not usually a 'short' note. It often happens at around F~349 Hz; the first fret on the open E. Ironically, the high E can't drive it as well as the Fs on the low E or D strings. Those Fs are lower in pitch, but the tension is changing twice per cycle, or at multiples of that, and they can rock the bridge by pulling on the top of the saddle in step with the motion of the top. The F on the high E pulls it too often. Since the bridge and top don't move much in that 'rocking' direction this doesn't result in 'short' notes (usually) but can 'enrich' the timbre of those notes. There are also some more complicated things that can go on with this one.
And so it goes. IMO it's precisely the complex nature of the way the guitar responds that makes it the instrument that we love. At the same time, the more effort we put into making it responsive and powerful, the more likely it is to cross over into behavior that is problematic. We're always trying to find that balance point where the 'great' instruments are. It's tricky.