Q: my iphone5 won't let me download ios 9.3 won't verify saying I have no internet connection when I have a full service

Many users have encountered problems updating their devices, activating them or signing it to their account after iOS 9.3 update. Apple is aware of the problem and is working on a fix.

Take a look at these articles to get additional information on the issue or fit it:

If you can't activate your iPhone, iPad, or iPod touch after installing iOS 9.3  - Apple Support

If you can’t activate your iPad 2 (GSM model) after you update to iOS 9.3If you can’t activate your iPad 2 (GSM model) after you update to iOS 9.3

Apple halts iOS 9.3 updates for older iDevices due to growing Activation Server Lock issues

Read this article: If you can't update to iOS 9.3 on your iPhone 5s or earlier, iPad Air or earlier, or iPod touch - Apple Support