It sounds as if one NIC is proxying for the IP other.
You may need to disable proxy arp on both interfaces. This is usually the default, but you can force it by making the proxy_arp files have a single "0" inside them instead of a "1".
echo 0 > /proc/sys/net/ipv4/conf/eth0/proxy_arp
echo 0 > /proc/sys/net/ipv4/conf/eth1/proxy_arp
You may also need to clear the arp tables of all the other servers and routers on the network.