Q: iOS 9.3 won't install

Try again later. Apple  servers may be overloaded. In case any further update issue occurs check the info on these links

Resolve iOS update and restore errors in iTunes - Apple Support

Get help with iOS update and restore errors - Apple Support

Useful pages but, my error does not have a numeric code.  The update has downloaded, but during verification I am told I am no longer connected to the internet — but I am.

This has happened to me yesterday and today.  I am able to update apps, so I know there isn't a problem with connecting to Apple's servers.  In fact I have updated my iPad successfully, but my unmodified iPhone will not take it.

iPhone 5S

Apple has pulled the iOS 9.3 update for older devices (iPad Airs and previous, iPhone 5s and previous) due to problems with activation. See these articles for details and possible workarounds.

http://www.imore.com/apple-working-ios-93-fix-older-iphone-ipad

https://support.apple.com/en-us/HT206203

Ah, thank you for pointing that out.  I wish the alert message simply said that.  It hadn't occurred to me the 5S might be considered "old" at this point, but I suppose it is. 

I agree...