Next time you remove the tube that will not stay inflated, match up the location of the hole in the tube with both the tire and rim. Then you will know where to look on the rim and tire for something that might be causing the hole. Is there something in the tire? Is there a burr on the rim or a spoke exposed at that point. When I install a tire or change a tube I always line the Name embossed on the tire with the valve hole. That way I will know exactly where to look for a thorn, piece of wire, or glass in the tire by where the hole is in the tube. Worn tires don't usually contribute to a tube failing unless there is a hole in the sidewall and I would expect you could see that right off.

Are you using tire irons to remove and install the tube? The best way to put the tire back on is with your fingers if that will work instead of using a tool. I never poked a hole in a tube with my thumb but sure have done so with a screwdriver (as a kid).