Many answers are negative assuming decline, or provide an extremely complicated answer to a simple problem. I'd propose based on decades of observation that you model the behavior and personality of all of management as if it was one person. Now is that imaginary merged individual person a lunatic deep in cocaine addiction psychosis? That might be a bad sign. Is that imaginary merged person a reasonable good leader?…
The problem I see with this advice is that many technical people (e.g. presumably many HN readers) are not good at "reading people". It's very similar to socially-inept engineers asking "how can I tell if a girl would be ok with me asking her out on a date?" We then respond with more well-meaning advice, "if she smiles at you, that's a good sign." And then the engineer might ask, "well, how can I tell if her smile is genuine affection or just politeness?" ... and so on and so on ... into a sort of infinite recursion into the mysteries of "people behavior".
If the engineer is the kind of person that can read Ribbonfarm's Gervais Principle[1] and say to him/herself, "that was a waste of time because that's all obvious information", then yes, he can follow your advice. He/she can evaluate the managers as if they were chess pieces and figure out the health of the company.
If you're the type that's not "tuned in" into how people exhibit implicit behaviors, you'll need a Plan B. E.g. look at the financials or other strategy.