Another Lee neck-sizer...
I too had problems w/neck tension w/certain bullets. Really didn't want to mess with changing the diameter of the mandrel. However, in the course of checking out the possible factors in the problem, I checked the diameters of the bullets I wanted to use.
Sure enough, the bullets I'd been using successfully were a thousandth or so more in diameter than the bullets that weren't getting enough neck tension.
Having a drill press to hand, I chucked the mandrel in the press and ran the drill, then looped emery cloth around the spinning mandrel. This holds it steadier and does a smoother job than doing it freehand, or using a hand-held drill. As per Lee's instructions, I stopped now & again to check the diameter of the mandrel--you don't want to reduce it any more than "just enough."
Anyhow, took the mandrel down only a couple of thousandths, IIRC, and that did the job. Put die back together, neck-sized some cases, and Voila! the slightly smaller bullets seated nicely, with sufficient neck tension to hold them in place.
FWIW, the larger bullets still seat perfectly well. Their neck tension will be more, I'm sure, but it doesn't seem to be enough different to affect accuracy, so I'm happy.
Bottom line: Lee's suggestion for reducing the size of the mandrel to increase neck tension works fine. If you do it, go slow and check periodically. Oh, and you have to reduce the mandrel over its entire length--if you only reduce it at the upper end where it does its work, the larger diameter lower part will just re-expand the neck as it pulls out of the case. And if you can get the use of a drill press, or a lathe would work fine for this, too, it's not just an easy process, it's really easy.
Good luck!