Are problems in NP decidable in polynomial time by NDTMs, or just recognisable? So if we have a problem in NP, does that mean that we can create, by definition, a NDTM that always accepts or rejects in polynomial time?  The alternative I'm thinking of would be if the NDTM could only be guaranteed to accept in polynomial time.  Rejecting could take longer or forever.
The decision problems studied in computational complexity theory are all decidable. One cannot define the worst-case time complexity for Turing machines that may fail to halt.