I've seen that problem manifest itself when setting up new user accounts in AD and copying a roaming profile "template" to the new user account. This was caused because I didn't have the "Everyone" group selected as having permissions to access the user profile.
 
I fixed this by logging in with a domain admin account, and changing the user permissions on the roaming profile.
 
Let me know if you need more info, and good luck

I guess what I meant is that if they have normal user rights their profile doesn't load, if I give them higher rights their profile loads.

Can you log in with Domain Admin rights (yourself, not the user account in question) and try the "copy to..." option, to ensure that the user has rights to his/her profile?

I had similar problem. What worked for me: INTERNAL permissions inside ntuser.dat were set incorrectly. Even though the users name and password on the new machines were the same as the old machines the unique identifier was different. Once I set permissions in each users .dat file for full access they could use their profile and update it etc. You can do this with regedt32 and load hive, set permission, then unload hive and changes are saved. The reason it works for you now when you log in as an administrator is that in ntuser.dat the administrator group always has access.