mysqldump question, now with complete sentences

[Madonna DeVaudreuil]

My sent folder shows that I sent  an email on this topic  before I 
finished editing it, but so far it hasn't shown up on the list.  If it 
does, my apologies.  Apparently I shouldn't have skipped watching the 
company training video on how to use our new email client!

We have a script that uses mysqldump to backup our databases to a network 
storage server.  Just recently this error started:

mysqldump: Error 2013: Lost connection to MySQL server during query when 
dumping table `xxx` at row: 1024037

I have had this problem before with two larger databases.  The solution 
was to back them up to local disk, and then copy the files to the storage 
server.  But I'd really like to know how to fix it,  because it's a pain 
to have to do this for each database.  I've googled the issue and found 
suggestions to increase max_allowed_packet.  It was 256 MB (as passed in 
to mysqldump, it's 128MB in my.cnf), but I changed it to 1GB  and still 
have the problem.  I also have a different database that is both larger in 
total size, and also has a larger table that the table name that is 
mentioned in the error message. 

Any suggestions for how to make this work?  We're using mysql 5.0.45 on 
CentOs 4.4.    The options I'm using with mysqldump are:
--quote-names --opt --max_allowed_packet=1GB --master-data=2 
--single-transaction --flush-logs --routines

Thanks,

Donna