Who's 3 DNS IP's are assigned in your router? You don't by chance have MAC spoofing enabled do you (on the DPC3825)? Also, have you taken a look at the modems error logs for any outstanding oddities? Sorry for all the questions I just find this behavior rather bizarre.
I've also read somewhere that having "Block IP Flood Detection" enabled in the DPC3825's configuration can cause random issues as well.