My Snap On box came with a strange, thin, almost paper-like liner - it's ike it's really compressed foam, but really thin. Kind of hard to describe. It seems to be hard wearing, and oil/water resistant, and doesn't tear when I chuck tools in or grab them out. These I've left in the two rows of small drawers, but in the full width drawers, and lift out tray, I cut some rubber matting up, that came on the shelf racking of one of my old vans, to fit. It's textured on one side, rather than smooth, and the other side (the side that is usually facing up) it's ribbed. I have it in reversed, so the ribbed side is facing down. I'd say it's 2, maybe 3mm, thick, so doesn't take up too much height. There's just room in the shallow drawers for a 32mm socket to lay on it's side and just clear the bottom of the drawer above it.
Anywayyyyyyyyy..... I was thinking about swapping it all out to matching liners, maybe a thin neoprene type material? Looking around on ebay and the like for universal liners, to cut to fit, and lots of boxes on here, seems to mostly be the anti-slip webbing type stuff. Not sure that's what I'm after - I think I'd prefer a "solid" (as in no see through gaps) liner, not too soft it's bouncy, but not so hard it's actually solid, if you follow me. And not too thick either - don't want to lost height as someone else mentioned. 2-3mm is probably the max I'd want to go to.