Q: Is it possible to backup regular text messages from phone to icloud?

Yes.

http://support.apple.com/kb/PH2584

That is a very generic reply with no specific information about the question originally asked.  When you go to configure iCloud backup, you can't select Text Messages as an option to be backed up. 

So, to ask the question again, how do you specifically back up regular text messages (not iMessages) on an iPhone (4S, iOS 7 or 8) from your phone to iCloud so you can put those old regular text messages on a new iPhone after you purchase one (like the iPhone 6)? 

Yes, unless you are restoring to another kind of device.

Note you cannot do selective backup, it's all or nothing.

Read here: About backups in iCloud and iTunes - Apple Support