consider yourself lucky
Mine does the same thing, but one tire always stops at exactly 12 PSI. I get a good two days out of an inflation and then I'm dragging out the compressor.
There are a couple things you can check. Get a spray bottle of someting sudsy and spray the valve and the entire inner and outer perimeter of the tire where it meets the wheel. If its leaking at the rim, you can have it dismounted and remounted using a silicone-type sealant at the bead. That usually solves the problem.
If its the valve stem leaking, you can unscrew the valve from the stem and replace it.
My guess is that you have aluminum wheels. They corrode around the bead and tend to let tiny leaks sneak by the tire. The best cure for this is to pull the tire off and use a wire wheel on a grinder to re-smooth the sealing surface. I would also suggest using the sealant when they're re-installed to be sure.
The reason it goes so far and then quits can be demonstrated by holding your breath. If you hold your breath by closing your lips and then try to push the air out, your lips yield letting air out. They are clamped together by a certain physical strength. In your tire anything over 28 psi will overcome the seal. It leaks until its down to 28, then it quits.