Patient is a virtue I am trying to work on No, I do honestly understand that these things can take years but I suppose I am just asking what happens if things like DNA are used up, or if you get too "stuck" in genealogical research - basically, does it always get solved? It's such a fascinating thing to think that ANY of these sorts of cases get solved, after so many years.
To start off: I’m no expert, and this is just my opinion.
Theoretically, the DNA could get used up, so that it was not possible to progress. In practice, the science keeps progressing. Things are possible today that would have seemed impossible just a few years ago.
One potential impossibility would be that records were lost and they can’t find where a person was buried. That’s happened. Or, some jurisdictions cremated their unidentified. That’s happened. Science can’t create something out of nothing.
Some genealogical situations are extremely difficult, but I don’t think there’s such a thing as being permanently stuck. At any moment someone new could add DNA that opened up new lines to research. However, as you said, patience is a virtue. The DNA Doe project recently announced an identification that they had been working on for three years—just the genealogy part.