Losing File Handles - But I am not using Files

Fellow Pascalians,
I am re-asking this as there has been no response. Perhaps my previous
header was to specific .......

My section has inherited an old DOS  TP program that handles comms that
is now giving us a lot of grief.

Since running under Win3.1 it has been ocassionaly failing with a runtime
209 - Lost File Handle. But at the point it fails we haven't opened or
closed any files other than an overlay file.

However recently it has been failing much more
often (possibly because the desktop is getting really full ie @ 75%+
resource usage).

The program uses overlays in EMS. We have determined that if we remove
the EMS code, the program works, but this is not a full solution as the
program will not then run in our standard software configuration.

Given that this has been inherited it is possible that we missed have  
something in compiler switches/ parameters  etc, or a problem with
OvrEMSInit or (gasp!) - Windoze or EMM386.

We have considered making the program Protected mode, but it is so poorly
written and structured that a rewrite would probably be faster and we
would exhaust the budget before we were anywher near complete.

All suggestions gratefully taken on board.