I cannot go 30 miles without a flat. I can't figure out what the problem is
I've had this exact problem.  I spent a whole lot of time removing the inner tube, patching it, etc...   All to no avail. I finally figured out that the side of a drilled spoke access hole was barely visible next to the rim tape.  When I'd inflate the inner tube, it would jut into that hole (depression) and get cut on the side of the drilled hole at the aluminum rim.  The fix was to get better (wider) rim tape, and carefully adjust it so there was a uniform overlap at the spoke access holes. 

I also figured out that is VERY difficult to patch a hole on the inner side of the tube.  You have to use a patch that is at least two times its width (long axis oriented along the rim...)

Good luck with it.  

Oh, and to help me in troubleshooting, I always label the fill port on the outside side of the tire (In my case I rotate the tire so the pressure rating is next to the fill valve. Then I use a paint pen to add a tick mark on the sidewall that exactly lines up with fill valve.)  This makes the pressure numbers easy to find, AND really help me in troubleshooting a leaky inner tube.  I can inflate the leaky tube in a bucket of water, find the leak, then remove it to line it up with both the rim AND the tire to search out root causes for the leak. 

Edit..
I'm guessing that with certain rim cross sections and geometry, and with a soft rim tape, it might be possible for no sharp edges be exposed with tire and tube is off of the rim, but under high tire pressure, the rim tape may move a bit, exposing the inner tube to the sharp side of a drilled spoke access hole, or the end of a long spoke...