When a ChromeCast connects to a new network, or reconnects to an existing, the device checks to see if it has an available firmware update from Google. If there is an update available, this is where the issue will occur. Customers will receive a notification in their ChromeCast app, required to connect the device, saying that the update has taken too long and failed.
This issue only occurs when initially connecting a ChromeCast that requires a firmware update. In most cases the device will connect with no issue.
Despite the failure message in Google's app, the update on the Chromecast will still proceed in the background. If you wait 10 minutes or so, the device should reboot itself once the update finishes and have no continuing issues.
This is a known issue with Chromecast when initially connecting to certain routers.