not able to or not allowed to

It is used, particularly in conversation, but it is not strictly accurate. "Not able to" is understood to include "not allowed to" informally but should not be used in that way in formal writing. "Unable to", on the other hand is rarely, if ever, used to mean "not permitted to", perhaps because it is considered to be rather more formal anyway.

Another term with the same sort of ambiguity is "can't" (and its opposite "can"). The meaning of these has become even more blurred than "not able" and "able". You will hear "Can I" being used when asking permission (as in "Can I borrow the car, Dad?") to the point where the more correct "may I" has almost disappeared from idiomatic use and sounds slightly odd in normal conversation.