Unable to login DBA in client with Borland $2501 error (also "TSysDBOjb failed system database connection validation.GetDBLIst";  error)

This error means your PC is running out of memory for BDE usage. (which is the Borland $2501 error). What you can try is to increase the amount of memory available to BDE, by doing the following.  

Go to the Control Panel (in Windows, not DBA), then into the BDE Administrator. Click on the "Configuration" tab, then click to expand the "System" icon below. You should see an entry for "INIT". Click on that, and values will appear in the right hand pane.

What you will then need to do is change the SharedMemSize entry from the default of 2048 to a larger multiple, such as 4096 or 8192 (see screenshot below for reference).

BDE memory-related errors (general info):

Occasionally, though usually not consistently, a computer may experience BDE memory-related errors. Usually, there are easy steps to resolve the error, and it generally doesn't resurface.

The BDE, currently being used by DBA, is somewhat older technology. It's very memory-concerned because memory resources, around its time of development, were much more critical/limited. Since then, memory limitations are generally more rare. We have been exploring other installation options that can eventually serve to replace this older BDE technology, and that can handle newer principles for memory handling/allocation. For now, though, we just have to deal with these occasional, rare errors.