Many Connection Errors

I have an MFC application that uses libmySQL.lib to access a remote database.  Within this application there is a thread that runs queries to produce live updates and there is also a facility to produce ad-hoc reports, so potentially there could be two queries being sent with two different connection handlers.

There is also a C++ application that runs on the remote database, writing insert queries to it on behalf of about 20 other machines at distance; i.e. these 20 machines do not talk to MySQL directly.

This system used to work fine until recently when my MFC application suddently said that it could not connect to the database.  I then used the lower-level MySQL command prompt to access the database and the following error message was displayed:

ERROR 1129: Host 'host81-136-213-32.in-addr.btopenworld.com' is blocked because of many connection errors.  Unblock with 'mysqladmin flush-hosts'

We arranged for this command to be executed on the remote server and it fixed the problem, but now we are finding it is happening with alarming regularity.  There are two possible reasons for this we can think of;

1. It has been gradually happening more frequently as more machines are added (i.e. the ones that do not communicate with the database directly, but via the server application).

2. We think, although we can't be sure, that the problems only started occuring after a Norton Firewall was installed on the remote server.

We have also noticed that the above connection error occurs on some PC's, while others in our building can still access the remote server without any problems at all (pointing to a client fault?)

We have not been using MySQL for very long and therefore don't know much about it's internal mechanisms.  Any pointers as to what could be causing this problem, and/or possible solutions would be greatly appreciated.