As you say probably more info is needed. There are numerous causes of this error. It is a 'catch all' type error which means 'I cannot communicate with the oracle process'. This usually occurs when the Oracle server process has died for some reason. But the oracle server does not go down does it?
If the failure is only on the client side you should focus on SQLNET and network issues.

Does this occur when trying to (a) make a connection to the database or (b) on an established connection ?

You migth want to try to do an ODBC trace.

At any rate it sounds like a network hardware problem. I.E. shitty network card
or a SQLNET configuration issue.

You migth want to compare your SQLNET.ORA file between computers that fail and those that don't for differences.

Have you got access to http://metalink.oracle.com/ ? There is a bunch of posts there on this.

What ODBC driver are you using Oracles? Microsoft's works best with VFP.

What version of oracle are you using, platform? 7.3.4.3
has a bug in it.