Is the physical connection dropping? Meaning, do the ethernet status lights go out, or do they show connectivity still exists?
Some things I have had to do with weird issues:
1) Set the switch port the 3300 is connected to 100 or 1000 full duplex, then set the LAN port in the controller the same (do not use Auto Mode)
2) Verify that any VLAN tagging is being done at the switch level and not at the 3300 (if possible), or verify that the correct Voice and Data VLAN are configured in the 3300
3) Replace the patch cord to the 3300 with a new, pre-molded on (NEVER use a home made patch cord to the 3300)
4) Change switch ports that the 3300 is connected to
5) Although this doesn't seem relevant, database corruption can cause very unusual issues sometimes. Do a full backup of the system and then a restore or even a re-installation of system software (full backup, manual re-install, apply licensing, restore)
I have also seen a controller with a bad ethernet chipset, causing weird issues... but replacing the chassis (if you don't have a spare one) is a last resort.
Oh, and you could also do a wireshark capture of a port mirrored to the port that is connected to the 3300 and break the packets down and see of there is reason the traffic isn't flowing anymore, you might find an incorrect VLAN tag, IP address, or who knows what.
