Yes, the account you use in CUCM to communicate with your AD server must have, at a minimum, read-only access to the LDAP database. It might be worthwhile to do a test sync with an AD-full-admin account configured in CUCM to see if the problem is related to the permissions or to your LDAP configuration in CUCM.
Also, you can run a trace in RTMT on the DirSync service and watch the interaction between CUCM and AD to see if that provides clues to the problem.
Here is a link to the chapter in the 11.5 system admin guide dealing with LDAP integration. LDAP integration hasn't changed so if you are running a different version the information will still be applicable.
CUCM System Configuration Guide - Chapter: Import Users From LDAP Directory
Let us know what you find out.
Maren