In 2003, we arrived at the BCV for two weeks on July 5th. On the 6th, 7th, and 8th, Epcot was astonishingly empty. Not deserted, but blissfully free of crowds. Relatively short waits for everything. MK was busier, but not terrible. I mentioned it to one of the CMs, who said, "Oh yeah, it's always like this after the 4th. Everyone clears out and there's a real lull. It'll pick up next week." And it did.
This year may be different, because of all the special packages Disney is offering (free dining, etc.) But I think it's safe to say July 6th will be more manageable than July 4th.
I'm hoping so anyway. We're arriving on the 2nd, and staying until the 21st.