Why are our iPhones such a POS?

What do you see in Settings>iCloud>Storage & Backup?

If that's showing as a small amount available, that is the source of your issue. If you're backing up to your computer, you can turn off iCloud backup; if you are not, tap on Manage Storage there and delete items as needed.

~Lyssa

Unless you purchased more space, there is only 5GB storage space in iCloud, regardless of the phone's storage capacity.

If you are backing up to the computer, then yes, tap on Manage Storage and delete a few items there. You don't have to delete everything, but you certainly can.

~Lyssa

Connect your phone to the computer you're using to backup.

In iTunes, click on the phone in the sidebar, then look at the Summary tab.

How much space is occupied by Other?

If this is more than 1-2GB, it could mean there are some corrupt files on your phone. Restoring should resolve the issue.

Try restoring from a backup and, if that does not help, you will have to restore as new.

~Lyssa

And how much of that is shown to be in "Other"?

That is way to much, you have corrupted files on your iPhone. Do a Restore from backup and see if it reduces that. If it does not you will need to Restore as a new iPhone and manually reload everything.