Arggghh, sorry guys. Forgot about the 1.6x factor in 720p (i.e. 60p) mode. Be that as it may, (1920x508x14bits/pixel)/8 bits/byte means 102,412,800 B/s, or 97.66MB/s. This is probably right around the limit for your card. Because the card isn't quite fast enough to keep up with the data rate (or not even really close, I don't know what your benchmark is), the buffer fills up very quickly and you start dropping frames after only 5 seconds. If you have it set to stop on dropped frames, then that's what happens: it stops.
To test, drop down to a lower total resolution (start much lower) and see if that works, then work your way back up. Where you have a problem, that's probably the limit of your card. But even if you can't shoot 1920x508, a 5dMkIII with practically no aliasing should allow you to shoot at 1728x456 or whatever the next lower resolution at 2.35 (in 60p) is. If you blow it up now to 1920, it should still look pretty damned good. Although there is apparently MORE aliasing (because of the additional 1.6 stretch) when shooting 60p. You're probably going to have to do some thorough testing.
Cheers!