I'll apply the same caveat as Mr. Geerkens, and go beyond that to acknowledge that this is a bad answer because it involves no sources. This is a frame challenge.
Historians are deeply interested in questions of “who” did what?
I'm skeptical. Both the Annales school and the Marxists (Arguably the Marxists are eisegetical (thank you for the new word)) emphasize the influence of abstract forces on history over that of "agents". In my opinion, "What happened?" is more important than "who did what?" (I'm aware that I've quoted imprecisely; after debugging excel all day, I'm not up to the challenge of finding the stack markup language to represent enclosed quotations. I don't think it damages the meaning).
What are the limits? The same as any other observational science. Based on the evidence, construct a narrative. (hypothesis or model). Test that narrative against other evidence. Does this narrative explain? If it were true, what other predictions could it make? If it were false, where would it conflict with existing evidence? How does this new narrative compare with the existing narrative? Where they differ, which has greater explanatory or predictive power?
If I recall correctly, the Whig narrative of history was undermined by Dr. Bailyn's research into revolutionary and federalist era pamphlets. The new evidence didn't support the established narrative and a new narrative was needed. Beard's narrative of Whig agency was diminished and a new narrative gave greater agency to socio-economic factors other than class. (my apologies for a crap summary of generations of scholarship.
What limits a historian? Plausibility. Predictive power. And of course, since it is science, the plausibility of competing theories.
Mr. Geerkens asked me to clarify predictive models in historiography. I avoided that in my original draft because I tend to rant and rave on the subject..... I will try to limit my rabid frothing, but I must confess it is something I don't articulate well because I get cranky. I'm also woefully out of date and I'm not sure where my basic reference texts are.
I'm a certified Risk Management Professional; there is a fairly common accusation against our profession that risk management and Bayesian statistics are not science because there is no control group, and we don't perform formal experiments. The rant usually starts when I ask where the control group for astronomy is. History is an observational science. Like astronomy or sociology, or economics or like Jane Goodall, we don't work with control groups - we do a form of regression theory - build a model, then test it against observations, and then test it again against other observations where it ought to be predictive. We tend to think about science as experimental, but there are branches of science that are purely observational.
I wish I remembered enough of the controversy between Beard and Bailyn to extend my discussion of Whiggish history above, but the example that comes to mind is Levi-Strauss' analysis of masks. (Lévi-Strauss, Claude. La Voie des masques (1972, The Way of the Masks, trans. Sylvia Modelski, 1982)). Strauss analyzed masks, and noted that certain features of a mask were associated with specific mythological patterns. (I haven't read this in at least 15 years, so I may have the details wrong; the work is available online, but behind a transaction wall) Masks with square eyes and dark colors were associated with one set of archetypes. If that association were valid, then we would expect (predict) masks with round eyes and light colors to be associated with complementary archteypes. This was true for the masks in question.
Another model is class structure - If Marxist theory were valid, then we would expect class identity to have more force than ethnic identity. Observationally, that is not true; the model does not predict behavior. Predictions of behavior based on the assumption of the proletariat are much less accurate than predictions of behavior based on perceptions of race. (I freely grant that upper socioeconmic classes actively interposed race as a tool to split the proletariat, but that tactic was more successful than any predictive model.
There has been a historical narrative that war is a male activity and soldiers were male. Scally LLama articulates an alternative model that explains more of the evidence we see in history.
In order to maintain the Beard model of Whiggish history, or the Marxist model of class, you need to encumber the model with ornate assumptions that become more and more cumbersome. Models that are simpler, and more efficient at prediction win.
I'm sorry that I'm not able to articulate those examples well - I'll look for better examples.