If I purchase additional storage, will I be able to log in (finally) to an iPhone giving me the "maximum number of free accounts" message?

The title says pretty much all. I have an iPhone which has reached the maximum number of free accounts due to an error on my part. It's currently logged in with my main AppleID/iCloud account, but any attempt to log out and log back in with the secondary account presents either the dreaded "Maximum number of free accounts" message, or simply "Could not log in".

All of the suggestions thus far to "just create another account on a different device or at icloud.com or on your computer" have proven to be useless. Any attempt to log in with anything other than the last three accounts associated with this device results in the one of these two error messages.

Yes, I have reset the phone. Yes, I have restored the phone. Multiple times. Yes, I can sign in separately to iCloud and the iTunes & App store using the second AppleID. But, the phone itself refuses to accept it.

Any suggestions or links to knowledge base articles that might answer the question posed in the title would be appreciated.