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?.