Whether or not an aircraft can depart from the intersection does indeed depend on all sorts of things: take-off weight, runway state (dry/wet and thus stopping distance), temperature, company policy, Captain's preference, etc etc. Once the factors have been looked up (originally in charts, now probably done by an iPad
), intersection-able or full-length will be revealed. This is all assuming that the operating company has the figures for the airport in question. Some don't.
There will always be a mix of some that can, some that can't. Whilst intersection take-offs are expeditious and good for runway utilisation, there's also the old adage that there's nuffin' as useless as runway behind you...