Both the Echo and the Home have error messages that handle when things don’t work. “Sorry, I’m not sure how to handle that yet” is what the Home says when it can’t get an answer. “Hmm… I’m not sure what you meant by that question” is what the Echo says.
In terms of service outage, the Echo is evoke a orange / yellow ring when there are connectivity issues. For the Home, it’ll say “Sorry, something went wrong.”
What’s missing is the potential for either device to do local network diagnostics. “Alexa, how’s my Internet connection?” or “OK Google, what’s my ping time?” would be fantastic tools.
It’s unclear whether Google or Amazon would look at adding this as a feature. It would require local based ASR or a fallback mode if Internet connectivity goes down.