Braid comes in different widths. If you use too wide of a braid for the power of the soldering pencil you are using, the braid will conduct and radiate the heat away so that it can never get hot enough to melt and absorb the solder.
Of course, if you're using an expensive temperature controlled soldering station, this is less likely to be a problem.
Also, even with the width of the braid well matched to the power of my pencil, I've had trouble with most brands of braid just not working well for some reason. I feel braid impaired. The stuff they sell at Radio Shack (Easy Braid?) has never worked well for me.
However, I have found that the Chemtronics brand of desolder braid works very well for me. I don't know why the brand of braid should make any difference, but it does for me. Perhaps the Chemtronics is impregnated with a different forumuation of flux? Anyway, now days I mail order the stuff from Digi-Key in the 25 ft. roles and keep an eye on my supply so that when I make a Digi-Key order I always check whether I need to refresh my supply of braid.