It is indeed quite a while since I had to cope with BDE problems, so my suggestion might be pretty outdated now. Nevertheless you can give it a try as it is only a small change in the registry.
What helped us in the past was disabling opportunistic locking at the server side (where the database files reside). For this you have to change the value of HKEY_LOCAL_MACHINE\SYSTEM\CurrentControlSet\Services\LanmanServer\Parameters\EnableOplocks from 1 to 0 (as 1 is the default value, it might even be absent). Note that you have to restart Windows to make this change effective.
A drawback of this setting is a possible overall network performance drop for that server system.