I hear/read this so often that I have to wonder in what circumstances the 2x extender produces good results, with which lens? Any?
What a teleconverter or extender does is magnify the centre of the image projected by the mother lens, so ultimate image quality depends on the sharpness of the mother lens, as well as the quality of the TC. The magnification causes a light loss of one stop with 1.4x and two stops with 2x (inverse square law).
In practise, it seems that a few high quality long zooms and primes can get away with the 1.4x with acceptable results, and the loss of only one stop of light is still practical. 2x TCs ask a heck of a lot from the mother lens, as well as pushing themselves further optically and the result is usually only good enough for critical work with high end primes.
One combo that a few people use is Canon 300L 2.8 with 2x TC, to make a 600 5.6 that is relatively affordable, still sharp, retains AF, compact, has IS etc. TBH I can't think of another combo for regular use; for occasional/emergency use yes, give a TC a try but if it really was a 'good' solution nobody would ever bother to spent several thousand quid on a long prime.
Results are basically down to the quality of the prime, plus the fact that TCs work better with longer lenses (200mm plus) that project a narrower light cone which is easier to handle optically.
There are a lot of MTF sharpness graphs in Canon Lens Work III, with and without extenders. The PDF is here, pages 23/24
http://software.canon-europe.com/files/documents/EF_Lens_Work_Book_10_EN.pdf