Why should we believe that $NEXP \not \subset P/poly$ I am sorry if this is not an advanced question. Most computer scientists believed that $NEXP \not \subset P/poly$ but they are not even close to this assumption. The main evidence that they are used is derandomization and they believe that $P=BPP$ and I know Nissan and Wigderson's generator which exist if $EXP \not \subset P/poly$($E \not \subset Size(2^{o(n))}$). On the other hand, I see some theorems like IP = PSPACE which thought to be false. Recently I read the IKW and there is a theorem states that $NEXP \in P/poly$ then there is a polynomial witness description for any language in $NEXP$. For me, it is likely to happen for example Succinct-HC is an $NEXP$-Complete language and it is likely to have a succinct witness. On the other hand, there are undecidable problems in P/poly that we don't know, maybe we could use them as an oracle to solve Succicnt-HC, They are some reasons in IKW‌'s paper but I need more references that help me to understand why should we believe that $NEXP \not \subset P/poly$.
The best evidence is in my opinion follows due to the results of Ryan Williams on even a mild speed up of $CIRCUITSAT$ provides $NQP\not\subset P/poly$ which is an extremely strong result compared to $NEXP\not\subset P/poly$. It indicates to me that either we are missing something trivial which would separate $NEXP$ from $P/poly$ or (remotely plausibly) anything that separates $NEXP$ from $P/poly$ would separate any class slightly bigger than $NP$ from $P/poly$.
Update It seems all the more likely if we do not reach $NEXP\not\subset P/poly$ by speeding up GapCircuitSAT or CircuitSAT problems we might achieve a separation via embedding $NP$ problems in $MCSP$ in $PTIME$ or $LOGSPACE$. It is unclear if speeding up $SAT$ has anything to do with embedding or vice versa. Please refer Comparing SAT to MCSP reduction class separations and faster SAT class separations?.