SQL Server does not exist or access denied

I'm getting occasional instances of the error:

[DBNETLIB][ConnectionOpen (Connect()).]SQL Server does not exist or
access denied

I get the error both in application code and from Enterprise Manager.
The database is LOCAL.

Microsoft SQL Server 2000 Developer Edition SP4
Windows XP Professional SP2

My application essentially reads one row from one table, reads some
additional rows from the another table in the same local database, and
then writes one row into another table in another local database (and
commits the transaction). The application process approximately 3932
input records, then the read for the additional rows fails with the
error above. With identical input data sometimes it will fail after
3931 rows, sometimes 3932, sometimes 3933. Once it fails, it will fail
for the next 5-6 input rows, and then start working for a few thousand
rows more, then error again.

Note that the application should not actually connect to the database
when I get this error, it's just opening a dataset on an already-open
connection.

During the time the application is getting errors (about 3 minutes
duration at a stretch), attempting to access the same local server with
Enterprise Manager gives the same "SQL Server does not exist or access
denied" error.

While things are failing, I observe that SQL Server processes are not
using any CPU (in task manager).

Things I have tried:
Disabling Norton Antivirus and Internet Worm Detection
Configuring SQL Server to use only one processor (it's a 2-Xeon
processor machine)

The only thing that seems to work is to set the connection timeout and
command timeout to 6 minutes.

.

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