Could LM317 in basic regulator configuration (see description) fail to regulate?
The cause MAY be inadequate minimum load current - see below.
Easily checked and easily fixed if this is what is wrong. 

As specified in the LM317 datasheet on page 3 and in figures 2, the LM317 needs a minimum output current to regulate correctly. This is 10 mA worst case at Vin-Vout = 40V differential, 3.5 mA typical. At 15V or so differential it drops to about 1.5 mA typical and one can guesstimate it may be around 4 or 5 mA wort case.

The output current is used to power the internal elecronics, unlike earlier generation device which uses Vin to Vref voltage for powering.

In your case with no external load Iout min is about Vref/Rref = 1.25V / 240 Ohms or about 5 mA. So you are on the upper bound of worst case so in mostinstances would expect to be OK - JUST. But, as a worst case spec is not given officially at the voltage it could be closer to 10 mA than I estimated.

Using a lower Rref or a permanent minimal load resistor will soon show if this is the cause of your problem.



ADDED

Connection seems a likely cause - see below.
Next most likely is a faulty IC. 

With Adj grounded (lh pin) and Vin on rh pin and say 100R from centre to lh pin you should see 1.24 ish Volts from centre to lh pins. 

Your 15.1V and 13.9V are 1.2V apart - which makes it sound as if you are measuring across the Vo-Vref pins.

Dim memory suggests that pinout MAY vary between TO220 and smaller pkg parts in some cases (pun noted). Check data sheet for actual part upou are using AND tell us what part is and give DS link. Standard pinout in almost all parts as per eg LM317 datasheet is left to right pins down, label side towards you AOI = Adjust Output Input .

Same for TO92 and TO89.

ie in your case
Vin on right hand pin
Two adjust resistors on left hand pin
Voltage out on centre pin.



Worth noting: On almost all regulators in 3 pin packages with a metallic tab the centre pin is the most negative voltage rise. So on an eg
7805 +5V regulator this is ground, on a
LM317 this is adjust and on a 
LM7905 -ve regulator this is Vin.
This allows substrate to be electrically connected to tab.
If the IC uses eg Silicon on Sapphire (you should be so lucky) all bets are off, but convention may be followed.