Here's what the support rep had me do. This first part applies for an iPhone with any carrier:
Go to Settings > General > Transfer or Reset iPhone > Reset, tap Reset Network Settings (please note this will removed any saved WiFi Passwords).
Once done, you should follow the steps below to remove any APNs and VPNs.
Go to Settings > General > VPN and device management > Delete any APN profile ( If you see a profile there )
Also Settings > VPN > Delete any existing VPN Profile
And Settings > Safari > Clear Cache/Data
The next step depends on your carrier, and you'll have to call their support for exact instructions. I'm on a carrier which uses the Verizon network, and these were the final steps for my phone:
After this, please follow the steps below to re-add the APNs.
Please open apn.droam.com in the Safari browser on your device. This website will help you set up APN settings for your device.
Select "Manually" from the options presented.
In the APN Name, enter "vzwinternet".
Touch the bottom-most empty box (which is actually the "Install Profile" button but you may not be able to see the label) to download the APN Profile.
Allow download for the profile.
After the profile is downloaded, go to Settings > General > VPN and Device Management > Select the downloaded profile > Install
Turn the airplane mode On for 30 seconds, then disable airplane mode.
Turn off WiFi and test the Cellular Data connection.