Well, the problem had only disappeared for a short time. If you manually send the credentials from the client code, this seems to work okay. Every time you instantiate a handler object, if you specify the credentials, the web server is happy. Like this:
using (OnTimeCustomerService.CustomerHandler onTimeServiceHandler = new OnTimeCustomerService.CustomerHandler())
{
onTimeServiceHandler.Credentials = System.Net.
CredentialCache.DefaultCredentials;
// ... perform actions.
}