One argument that I have come across surprisingly often in recent times is this: Many smart people have thought hard about topic X and found it difficult, therefore X is a real and difficult problem. Examples would include the hard problem of consciousness and Gettier problems.
A closely related and likewise frequently heard claim is that some person Y's work and opinion must be taken seriously because they know a lot about a topic, have given serious thought to it, and published many books on it; in this case I remember the claim being made with reference to some theologian who Richard Dawkins had interviewed and whom he had supposedly treated very dismissively and arrogantly, but I have not seen the interview myself, so I cannot judge.
The problem is that while it may well be true that topic X is a real and difficult problem, or that person Y should be taken seriously, neither conclusion follows from this kind of argument.
Focusing on the first claim, my favourite example is the exact nature of the trinity because very few people, even few of those calling themselves Christians, still take it seriously. But over centuries, thousands of scholars and theologians have hotly debated issues such as whether the Son is of the same Substance as the Father or whether the Son was subordinate to the Father, for example. Lots of ink and paper were used, meetings were held, people were excommunicated, and heretics were persecuted.
So clearly this issue is one that smart, educated people have thought about for a long time and found very difficult to resolve (unless you count resolution by force, which you shouldn't because when contemplating how difficult a problem is we are talking an intellectual solution). Does that mean that the problem is actually hard? Does it mean that the problem is a real problem in the first place?
Well, everybody is free to believe what they want, but my answer is: obviously not. There is no commonly agreed on evidence, no evidence that would convince a reasonable outsider, that any gods exist at all. Worse, there appear to be fairly good, positive arguments that the very specific god of medieval Christianity, as opposed to the lofty god of sophisticated contemporary Christians, cannot possibly exist. For starters, the trinity itself seems logically incoherent (which is probably its whole point), but even god's various attributes like omnipotence and omniscience are often self-contradictory and contradicting each other.
And of course it is a mere accident of history that we do not currently have thousands of theology lecturers at European and American universities discussing the Mithraic mysteries with their students instead of the trinity. If another cult had won the support of the Roman emperor at a crucial moment, it is quite likely that nobody today would consider the latter concept more than a weird footnote in the religious history of late antiquity.
In other words, to me at least the problem of the nature of the Son is a non-problem because I see no reason to assume that the Son and the Father exist in the first place. Yet it was considered to be a real and difficult problem for centuries by the brightest and most educated people of the time. And even those who disagree, even Christians who do believe that the trinity makes sense, could probably easily come up with their own examples of something similar happening; as Christians, they would at least have to think the same about the theological controversies in a competing religion.
That should caution us against accepting other issues as real and difficult just because smart and highly educated people say they are. There needs to be some additional, independent evidence on top of that.
Likewise, merely thinking hard and writing a lot about an issue does not mean that one deserves to be taken seriously. If the issue in question is founded on false assumptions, then all the effort was merely wasted, and the specialist is an expert on something that doesn't exist.