I believe that I have answered my own question.
I got to thinking that clicking on a RadToolBarButton is causing a postback which would cause the validator to fire and prevent the postback. This theory was borne out by the following thread I found on StackOverFlow:
RequiredFieldValidator + Button = Postback!
I resolved the issue by adding CausesValidation="false" to the RadToolBarButton tag.
- Robert Eberhart