Error BLZBNTBNA00000005 appears with the message that the Battle.net agent went to sleep, which means the background agent that handles downloads, updates, and logins lost its connection to Battle.net. Dismissing the message usually brings it straight back, because the agent itself is still failing to connect.
If the agent keeps going to sleep on every launch, uninstall Battle.net, delete the leftover Battle.net folders in ProgramData and %localappdata%, then install a fresh copy from Blizzard. If the error only appears on one network, try a different network, since a strict router or network filter can block the agent's connection.
What is the Battle.net agent? The agent is a background helper program that Battle.net uses for downloads, updates, and sign-in. When it loses its connection, Battle.net cannot manage games, and it shows this error.
Why does the error keep popping up after I dismiss it? Dismissing the message does not fix the agent, so it fails again a few seconds later and shows the message again. The agent has to be restarted or its cache cleared before the message stops.
Is BLZBNTBNA00000005 caused by my internet? It can be. A dropped connection, a firewall block, or a router filter can all cut the agent off. It can also be a Blizzard-side problem, so check the service status before changing local settings.