Is it possible to backup an iPhone to both iCloud and iTunes? This question is generic but for my case I have an iPhone SE with iOS 10.3.1 and a MacBook Pro running Sierra 10.12.4.

Is it possible to have my iPhone backup to both iTunes and iCloud? When I connect the phone to iTunes the backup option seems to have a radio option for one or the other.

But it seems it would be preferable to have backups in two different places, a more complete larger backup on the computer and a smaller backup to iCloud that has the settings/app list/documents.
Yes, this is possible.

The options you're referring to in iTunes relate to automatic backups. You can achieve what you want by enabling iCloud backups (either in iTunes or in your iPhone's settings) and then manually doing a backup to iTunes whenever you want. 

To do a manual backup to iTunes, simply connect the iPhone as you usually do with iTunes and then in the device's Summary screen within iTunes (the one with the backup options) you'll see to the right a Back Up Now button. Just click this whenever you want to also backup to iTunes.