We are getting a lot of reports coming in of the upgrade (!) to iOS9.3 failing, see screen shot attached.
It appears that this is not a new problem, a quick google search shows forums reporting this error for iOS7.x, basically it says wait for the update to update.
Other threads take you through flushing the device memory, toggling on and off the WiFi etc but the explanation from ComputerWorld appears the most sensible.
Quote from Computer World [sic]
Computerworld | Mar 28, 2016 1:09
Apple today replaced the iOS 9.3 update with a new version after the
original crippled some older devices last week, and confused large numbers of
unaffected customers when their iPhones and iPads told them that they weren’t
connected to the Internet when in truth they were.
The initial version of iOS 9.3, which Apple issued a week ago, threw owners
of older hardware – iPhone 5S and earlier, and iPad Air and earlier – for a
loop when it asked them for their Apple ID and password, normally not a step in
the iOS update process.
Apple published a support document that acknowledged the problem and
Last week, Apple pulled iOS 9.3 from its servers in an effort to prevent
more people from seeing the demand for an ID and password. Today, the
Cupertino, Calif. company re-released iOS 9.3.
In the meantime, however, users whose devices had already downloaded
iOS 9.3, but whom had not authorized the upgrade, began seeing a message when
they got around to updating. “Unable to Verify Update,” the on-screen
message read. “iOS 9.3 failed verification because you are no longer
connected to the Internet.”
But those devices – including an iPhone 6 owned by a Computerworld
staffer – were connected to the Internet.
A thread (https://discussions.apple.com/thread/7506344)
on Apple’s support forum spelled out the problem, and the likely cause.
“I suspect this is really Apple’s way of blocking you from installing
the update you have already downloaded, until they fix the known activation
issue with the update,” wrote someone identified as
"AndreTheGeek" on March 25. “Wait for the update to the update.”
Apple probably dropped or changed the digital “signature” of the
update on its end, creating a mismatch that resulted in the “Unable to
Verify Update” message.
Although the thread contained fewer than 30 message, it had been viewed more
than 10,000 times, a very large number for such a small collection of reports.
In other words, lots of users had seen the same no-Internet error, searched the
Web using a string such as “ios 9.3 verification failure,” as did Computerworld,
and were directed to Apple’s support forum.
Computerworld’s device successfully downloaded and installed the
revised iOS 9.3 today.