Why did Screen Sharing suddenly stop working (10.6.8)?
The last edit was the primary clue ... it was not a VNC-only issue, but a intermittent network issue.

We had a small 5-port hub in the room to support three machines off of the one network drop. Once we removed the hub and plugged the machine directly into the drop all was good. The hub had been there, trouble-free, for more than 3 years.

Once again I am reminded to ask myself more questions. An obvious question would have been: Does the target Mini have good connection to the network, outside of VNC/SSH? Is it dropping packets to reliable servers?