What to do if APN settings appear disabled
Sometimes, APN settings on your device for a particular carrier may be "locked" such that they are "grayed out" and can't be modified.
This is often an indication that they have been set by your currently connected carrier and that you shouldn't need to modify them. However, in some cases it is possible to work around this and gain access to modify them again.
Reset to default option
At the APNs screen, try the Reset to default option available by pressing the overflow menu button. This will delete any of your customizations to APN settings and return them to their initial state as provided by the manufacturer.
If your carrier has pushed APN settings to you, this may revert from those settings back to the ones included on your device, and get you back the ability to add new ones.
Airplane mode option
Another thing you can try is to remove the SIM card or enter Airplane mode (called "Flight mode" on some devices) prior to selecting the Reset to default option. If your carrier tries to push new APN settings to your device regularly, this may afford you enough time to perform your modifications before that happens again and you lose access to edit them.
Note that if you remove the SIM card, your device may need to be rebooted afterwards.
If you removed the SIM card, replacing it may require another reboot.