Are you using an ICD2 to program the PIC? If so, make sure to use the 
Program menu in MPLAB, not the Debug menu. If the chip is programmed 
with the debug executive, it will take control of RB6 & RB7 (program 
clock and data). Also make sure LVP is not enabled, that would disable RB5.



>What's really confusing me is that when RB7, RB6 and RB5 are high
>(still one at a time), my multimeter shows only ~0.6volts on those
>pins. When RB4 and RB3 are high, I get ~4.7 volts and those wires are
>tested accordingly.

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