Since this is working properly on all other computers except these 2, I would start by trying to determine what is different about these computers. OS, service pack, other software running (particularly AV software)? Next I would check if the ODBC's are set up differently from your other computers. Are you using a different version of the SQL driver? In general, I have found SQL Server to be more 'stable' than SQL Native Client, however, if you are using functionality that SQL Native Client is needed for, you may not be able to switch. Might be a good test, however, if only to rule it out.
Also, is there anything in the Windows event log of these 2 computers that would indicate any kind of network or connectivity issue?