Okay, in the end, here's what I did to fix the problem.

Set the Central to be my desired fixed address of 192.168.0.4
with a subnet mask of 255.255.0.0

Set my internet router to have a subnet mask of 255.255.255.0

Set the Central's "receivers" screen to make the receivers be in the address range of 192.168.1.142 through 149, with subnet masks of 255.255.0.0

This allows the receivers to get their DHCP requests ACK'd by the Central, while the internet router ignores them. The larger subnet allows the Central to continue to serve the receivers, even though the internet router doesn't talk to them. And the Central can still talk to the internet router and get to the internet for its gracenote lookups.

Yay!

Thanks for the tips, guys.

To the original poster of this thread: Hope this helps? Dunno whether you were running a Central or if you were running the ARM software. If you're running the ARM software, see if there's a setting that allows the receivers to get their DHCP addresses from your internet router instead of from ARM.