First quick thoughts, and please forgive if these are things you've already checked/considered:
1/ Pinout - are you absolutely sure the device is wired right? (I know I've blown, or at least shut-down, regulators before when prototyping)
2/ Is the LM317T getting hot - that's a lot of voltage for a linear device to be dropping? - it may be in thermal shutdown, and since it doesn't have a ground terminal, I'm not sure what'll you see if it can't cope.
3/ What values have you got for the output-to-adjust feedback resistor and the pot? and is it a lin or log pot?
4/ Is there any real reason to use an adjustable regulator? - a linear 7805 variant reg would be dissipating vast heat, but I make extensive use now of switching regulators such as Recom R-78C5.0 - they can run off up to about 40v input, but because they're switching devices, their efficiency is much much higher & they don't heat-up anything like so much
Edit: oops - too sloooooow a typist! - cross-posted same time as John & Andreas