Hi There
You would really need to post the "sanitized" configuration of your core switch(es) and at least 1 or 2 of your problem switches along with a description of how they are connected for anybody to be able to give meaningful advice.
Also your core switch the "(3500)", is this a layer 3 switch (3550/3560)? How many VLAN's are configured on your LAN? How is inter VLAN routing handled? If there is a router involved please post the "sanitized" configuration from this also
There are lots of possible causes. Are there ACL's configured? Are some of your switches "daisy chained" together where 1 switch is down which causes the downstream switches to be unavailable? Are there Routing problems? Do you have hardware issues on the ports on the core switch that connect to these offline 2950's etc.
Best Regards,
Michael