I'm programming using a PICkit2 and pk2cmd, with this line:
pk2cmd -P -I -M -FDIO_16F627A.hex
I've tried it with and without the -X option (VPP first), and with slower programming speeds, with the same result.
During programming, I'm disconnecting VCC from the rest of the circuit and letting the PK2 power the device.
It will be really great if I can get these things to program, mainly so I don't run into this same surprise again with another part. I'm about to order a tube of 16F887, though, just to be completely sure that I'll have something to finish my project with...