ooh another thought is autonegotiate. Setting auto for speed and particularly duplex on your internal NIC can cause problems, especially with NICs/Switches that don't autoneg well. Some cases this will cause the NIC or switch port to go to 10/half duplex, and under load will cause massive amounts of frame collisions which will effectively lock your port up. Nothing will be able to communicate with the core which means your MDs will die, but all other machines on that switch will be able to ping each other just fine.
The suggestion I made to connect direct to the core could eliminate this possibility, too. If your switch is manageable at all, you can force the switch port to 100/Full or 1G, and then do the same to the core NIC to prevent this. Also, if it is manageable, for a Cisco you can turn on PORTFAST which will eliminate certain STP blocking issues. Other brands of manageable switches will have an equivalent to this option as well. Can explain what this is if it turns out to be the cause.
But suggestions about the patch cables are the first and most likely causes, be sure of this first as it will be annoying for you to go through lots of advanced troubleshooting only to find you had a crappy cable