Certain problems are known to be undecidable, but it is nevertheless possible to make some progress on solving them. For example, the halting problem is undecidable, but practical progress can be made on creating tools for detecting potential infinite loops in your code. Tiling problems are often undecidable (e.g., does this polyomino tile some rectangle?) but again it is possible to advance the state of the art in this area.
What I am wondering is if there is any decent theoretical method of measuring progress on solving undecidable problems, that resembles the theoretical apparatus that has been developed for measuring progress on NP-hard problems. Or does it seem that we are stuck with ad hoc, I-know-progress-when-I-see-it assessments of how much particular breakthroughs advance our understanding of undecidable problems?
Edit: As I think about this question, it occurs to me that perhaps parameterized complexity may be relevant here. An undecidable problem may become decidable if we introduce a parameter and fix the value of the parameter. I'm not sure if this observation is of any use, though.