An old aviator's proverb states: "The three most useless things in aviation are altitude above you, runway behind you, and fuel left at the pump." Curmudgeonly advice aside, it is actually quite common to "leave runway behind you." Sit around most airports and you'll see all sorts of airplanes using less than full runway length for takeoff. The rationale behind intersection departures is simple: it saves time, both in taxiing and sometimes in waiting behind other aircraft.
I've known captains who categorically refuse to accept an intersection departure, ever. I've also seen aircraft depart from intersections under conditions I'd never accept. The prudent pilot, though, knows that intersection departures can be safely made under certain conditions, and weighs all the safety factors before deciding if this takeoff meets those conditions.
Personally, I'd go full length every single time if I was flying a light twin with marginal engine-out performance. In these airplanes, an engine failure just after takeoff is quite often fatal; only 1000' of extra pavement can make the difference between buying the farm and saving the day. An engine failure on takeoff in a single engine airplane is only half as likely and less prone to fatality, but I still go full length if there aren't suitable landing areas off the end of the runway.
My airline's MegaWhacker Flight Standards Manual says: "The [MegaWhacker] has certain performance characteristics that offer the flexibility and advantage of operating from runway intersections that are unsuitable for other aircraft. Substantial savings in time and fuel can be gained by using intersection departures. Therefore, if specific runway intersection departures are authorized in the performance, pilots are encouraged to use them."
At most of the airports we operate out of, performance almost always allows for departure from one intersection or another. If the paper performance was the only consideration, as our company's manual seems to argue, we'd do intersection departures all the time, no questions asked. The problem is, paper performance doesn't provide for very big margins, and it was computed using a brand new aircraft flown by a sharp test pilot. An intersection takeoff may still provide a huge margin over the minimum performance, or it may cut it very close. In the latter case, taking the time to taxi full length is cheap insurance. Many captains I fly with will refuse intersection departures with a wet or contaminated runway, or with a tailwind, or at heavy weights, even if the performance says we can do it.
Performance isn't the only consideration. At major airports, wake turbulence can be a big hazard, and intersection takeoffs increase the danger. When departing behind heavy aircraft, we can almost always outclimb their flight path, keeping the wake turbulence well below us. Departing from an intersection after a heavy, though, even the MegaWhacker may lift off further down the runway than the heavy did, meaning that there is a good chance of encountering heavy wake turbulence very early in the climb. Even smaller aircraft not classified as heavy can produce potentially dangerous wake turbulence. The MiniWhacker, for example, throws out a very impressive wake for it's small size.
So, when do I personally feel comfortable accepting or requesting an intersection departure? When the takeoff performance provides decent margins, when the aircraft isn't near max gross weight, when the weather is good, and when I don't expect wake turbulence to be a consideration. Any other conditions decrease safety unacceptably for a small time savings. Of course, accepting or declining an intersection departure is always the captain's call, but most captains will listen if their first officer would rather not take an intersection.