Are there circumstances where a claim has been twice rejected, where you cannot file an appeal?

Yes.  Another requirement for appeal is that the application be under rejection.  This means that Applicant cannot file a notice of appeal after filing a continuation application, but before receiving a rejection on that continuation application.  Applicant also could not file notice of appeal with a request for continued examination (RCE) -- even if the patent application had been twice rejected prior to the filing of the RCE.