How can you tell if a faculty or group is on decline? Am I being used?
In the final year of my PhD, I made a tour of several places in the USA to try to establish contacts in the hope of finding hidden postdoc opportunities.  I knew of those places because I frequently saw their papers.  In some places, I have seen signs of decline:


In one place, the main senior scientist had moved away some years ago.  Since then, one by one, others had left.  It appears nobody was replacing them.  The people who were still there appeared to be the ones who were less successful at getting grants and writing papers, than the ones who had left.
In another place, I didn't see any PhD students in a group, and only a handful of postdocs.  Most people were above 50, with the head of the group being well above 70.  Although certainly a great name to have as a support, I wouldn't recommend anyone starting a PhD project with someone who will turn 80 before the candidate is expected to get their PhD.  
I also overheard conversations from which it was clear that people were not very ambitious.  I almost literally heard someone say, that they now had a permanent position so they didn't need to write a lot of papers anymore.


Those are some signs that a place might be in decline: Have more than two senior scientists left the place in the past five years?  Is the median age above 60?  Are some people happier to stay on their desk than to attract great PhD students and postdocs to produce some great science?